Abstract Algebra€¦ · Chapter 1 Groups 1.1SEMIGROUPS, MONOIDS AND GROUPS Deﬁnition 1.1.1. If G...

Abstract Algebra

Algebra

Travis Dirle

December 4, 2016

Contents

1 Groups 11.1 Semigroups, Monoids and Groups . . . . . . . . . . . . . . . . 11.2 Homomorphisms and Subgroups . . . . . . . . . . . . . . . . . 21.3 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Cosets and Counting . . . . . . . . . . . . . . . . . . . . . . . 41.5 Normality, Quotient Groups, and Homomorphisms . . . . . . . 51.6 Symmetric, Alternating, and Dihedral Groups . . . . . . . . . . 61.7 Direct Products and Direct Sums . . . . . . . . . . . . . . . . . 7

2 The Structure of Groups 112.1 Free Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . 112.2 Finitely Generated Abelian Groups . . . . . . . . . . . . . . . . 122.3 The Krull-Schmidt Theorem . . . . . . . . . . . . . . . . . . . 132.4 The Action of a Group on a Set . . . . . . . . . . . . . . . . . . 142.5 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . 162.6 Classification of Finite Groups . . . . . . . . . . . . . . . . . . 172.7 Nilpotent and Solvable Groups . . . . . . . . . . . . . . . . . . 172.8 Normal and Subnormal Series . . . . . . . . . . . . . . . . . . 19

3 Rings 213.1 Rings and Homomorphisms . . . . . . . . . . . . . . . . . . . 213.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Factorization in Commutative Rings . . . . . . . . . . . . . . . 273.4 Rings of Quotients and Localization . . . . . . . . . . . . . . . 293.5 Rings of Polynomials and Formal Power Series . . . . . . . . . 313.6 Factorization in Polynomial Rings . . . . . . . . . . . . . . . . 33

4 Modules 374.1 Modules, Homomorphisms and Exact Sequences . . . . . . . . 374.2 Free Modules and Vector Spaces . . . . . . . . . . . . . . . . . 414.3 Projective and Injective Modules . . . . . . . . . . . . . . . . . 434.4 Hom and Duality . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6 Modules over a Principle Ideal Domain . . . . . . . . . . . . . 49

i

CONTENTS

4.7 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Fields and Galois Theory 535.1 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 The Fundemental Theorem . . . . . . . . . . . . . . . . . . . . 555.3 Splitting Fields, Algebraic Closure and Normality . . . . . . . . 575.4 The Galois Group of a Polynomial . . . . . . . . . . . . . . . . 595.5 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.6 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.7 Cyclic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 635.8 Cyclotomic Extensions . . . . . . . . . . . . . . . . . . . . . . 655.9 Radical Extensions . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Commutative Rings and Modules 676.1 Chain Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Prime and Primary Ideals . . . . . . . . . . . . . . . . . . . . . 686.3 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . 706.4 Noetherian Rings and Modules . . . . . . . . . . . . . . . . . . 716.5 Ring Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 726.6 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . 736.7 The Hilbert Nullstellensatz . . . . . . . . . . . . . . . . . . . . 74

7 Linear Algebra 777.1 Matrices and Maps . . . . . . . . . . . . . . . . . . . . . . . . 777.2 Rank and Equivalence . . . . . . . . . . . . . . . . . . . . . . . 797.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.4 Decomposition of a Single Linear Transformation and Similarity 847.5 The Characteristic Polynomial, Eigenvectors and Eigenvalues . . 87

ii

Chapter 1

Groups

1.1 SEMIGROUPS, MONOIDS AND GROUPS

Definition 1.1.1. If G is a nonempty set, a binary operation on G is a functionG×G→ G. A semigroup is a nonempty setG together with a binary operationon G which is

(i) associative: a(bc) = (ab)c for all a, b, c ∈ G; A monoid is a semigroup Gwhich contains a

(ii) (two-sided) identity element e ∈ G such that ae = ea = a for all a ∈ G.A group is a monoid G such that

(iii) for every a ∈ G there exists a (two-sided) inverse element a−1 ∈ G suchthat a−1a = aa−1 = e. A semigroup G is said to be abelian or commutative ifits binary operation is

(iv) commutative: ab = ba for all a, b ∈ G.

Theorem 1.1.2. If G is a monoid, then the identity element e is unique. If G is agroup, then (i) c ∈ G and cc = c⇒ c = e;

(ii) for all a, b, c ∈ G ab = ac⇒ b = c and ba = ca⇒ b = c (left and rightcancellation);

(iii) for each a ∈ G, the inverse element a−1 is unique;(iv) for each a ∈ G, (a−1)−1 = a;(v) for a, b ∈ G, (ab)−1 = b−1a−1;(vi) for a, b ∈ G the equations ax = b and ya = b have unique solutions in

G: x = a−1b and y = ba−1.

Theorem 1.1.3. Let R(∼) be an equivalence relation on a monoid G such thata1 ∼ a2 and b1 ∼ b2 imply a1b1 ∼ a2b2 for all ai, bi ∈ G. Then the set G/Rof all equivalence classes of G under R is a monoid under the binary operationdefined by ab = ab, where x denotes the equivalence class of x ∈ G. If G is an(abelian) group, then so is G/R.

Definition 1.1.4. Let G be a semigroup, a ∈ G and n ∈ N. The element an ∈ Gis defined to be the standard n product

∏ni=1 ai with ai = a for 1 ≤ i ≤ n. If G

CHAPTER 1. GROUPS

is a monoid, a0 is defined to be the identity element e. If G is a group, then foreach n ∈ N, a−n is defined to be (a−1)n ∈ G.

The order of a group G is the cardinal number |G|.

Theorem 1.1.5. If G is a group (resp. semigroup, monoid) and a ∈ G, then forall m,n ∈ Z:

(i) aman = am+n (additive notation: ma+ na = (m+ n)a);(ii) (am)n = amn (additive notation: n(ma) = mna).

If a group G is abelian, then (ab)n = anbn for all n ∈ Z and all a, b ∈ G.

1.2 HOMOMORPHISMS AND SUBGROUPS

Definition 1.2.1. Let G and H be semigroups. A function f : G → H is ahomomorphism provided

f(ab) = f(a)f(b), for all a, b ∈ G. (1.1)

If f is injective as a map of sets, f is said to be a monomorphism. If f is surjec-tive, f is called an epimorphism. If f is bijective, f is called an isomorphism.In this case G and H are said to be isomorphic (written G ∼= H). A homo-morphism f : G → G is called an endomorphism of G and an isomorphismf : G→ G is called an automorphism of G.

Definition 1.2.2. Let f : G → H be a homomorphism of groups. The kernelof f (denoted Ker f ) is {a ∈ G : f(a) = e ∈ H}. If A is a subset of G, thenf(A) = {b ∈ H : b = f(a) for some a ∈ A} is the image of A. f(G) is calledthe image of f and denoted Im f . If B is a subset of H, f−1(B) = {a ∈ G :f(a) ∈ B} is the inverse image of B.

Theorem 1.2.3. Let f : G→ H be a homomorphism of groups. Then(i) f is a monomorphism if and only if Ker f = {e};(ii) f is an isomorphism if and only if there is a homomorphism f−1 : H → G

such that ff−1 = 1H and f−1f = 1G.

Definition 1.2.4. Let G be a group and H a nonempty subset that is closed underthe product in G. If H is itself a group under the product in G, then H is said tobe a subgroup of G. This is denoted H < G.

Theorem 1.2.5. Let H be a nonempty subset of a group G. Then H is a subgroupof G if and only if ab−1 ∈ H for all a, b ∈ H .

Corollary 1.2.6. If G is a group and {Hi : i ∈ I} is a nonempty family ofsubgroups, then

⋂i∈I Hi is a subgroup of G.

2

CHAPTER 1. GROUPS

Definition 1.2.7. Let G be a group and X a subset of G. Let {Hi : i ∈ I bethe family of all subgroups of G which contain X. Then

⋂i∈I Hi is called the

subgroup of G generated by the set X and denoted 〈X〉.

Theorem 1.2.8. If G is a group and X is a nonempty subset of G, then thesubgroup 〈X〉 generated by X consists of all finite products an1

1 an22 · · · ant

t (ai ∈X;ni ∈ Z). In particular for every a ∈ G, 〈a〉 = {an : n ∈ Z}.

Remark 1.2.9. Composition of homomorphisms (resp. mono, epi, and iso-morphisms) of semigroups, is also a homomorphism. For a homomorphismof groups f : G → Hthen f(eG) = eH . Furthermore f(a−1) = f(a)−1 forall a ∈ G. Two examples of subgroups of a group G are G itself and the triv-ial subgroup 〈e〉. A subgroup H such that H 6= G, H 6= 〈e〉 is called a propersubgroup. If G = 〈a1, . . . , an〉, (ai ∈ G), G is said to be finitely generated.Ifa ∈ G, the subgroup 〈a〉 is called the cyclic (sub)group generated by a. If H andK are subgroups, the subgroup 〈H ∪ K〉 generated by H and K is called the joinof H and K and is denoted H ∨ K (additive notation: H + K).

1.3 CYCLIC GROUPS

Theorem 1.3.1. Every subgroup H of the additive group Z is cyclic. Either H= 〈0〉 or H = 〈m〉, where m is the least positive integer in H. If H 6= 〈0〉, then His infinite.

Theorem 1.3.2. Every infinite cyclic group is isomorphic to the additive groupZ and every finite cyclic group of order m is isomorphic to the additive groupZm.

Definition 1.3.3. Let G be a group and a ∈ G. The order of a is the order of thecyclic subgroup 〈a〉 and is denoted |a|.

Theorem 1.3.4. Let G be a group and a ∈ G. If a has infinite order, then(i) ak = e if and only if k = 0;(ii) the elements ak(k ∈ Z) are all distinct.

If a has finite order m > 0, then(iii) m is the least positive integer such that am = e;(iv) ak = e if and only if m|k;(v) ar = as if and only if r ≡ s( mod m);(vi) 〈a〉 consists of the distinct elements a, a2, . . . , am−1, am = e;(vii) for each k such that k|m, |ak| = m/k.

Theorem 1.3.5. Every homomorphic image and every subgroup of a cyclic groupG is cyclic. In particular, if H is a nontrivial subgroup of G = 〈a〉 and m is theleast positive integer such that am ∈ H then H = 〈am〉.

3

CHAPTER 1. GROUPS

Theorem 1.3.6. Let G = 〈a〉 be a cyclic group. If G is infinite, then a and a−1

are the only generators of G. If G is finite of order m, then ak is a generator ofG if and only if (k,m) = 1.

1.4 COSETS AND COUNTING

Definition 1.4.1. Let H be a subgroup of a group G and a, b ∈ G. a is rightcongruent to b modulo H, denoted a ≡r b(modH), if ab−1 ∈ H . a is leftcongruent to b modulo H, denoted a ≡l b(modH), if a−1b ∈ H .

Theorem 1.4.2. Let H be a subgroup of a group G.(i) Right (resp. left) congruence modulo H is an equivalence relation on G.(ii) The equivalence class of a ∈ G under right (resp. left) congruence mod-

ulo H is the set Ha = {ha : h ∈ H} (resp. aH = {ah : h ∈ H}).(iii) |Ha| = |H| = |aH| for all a ∈ G.

The set Ha is called a right coset of H in G and aH is called a left coset of Hin G. In general it is not the case that a right coset is also a left coset.

Corollary 1.4.3. Let H be a subgroup of a group G.(i) G is the union of the right (resp. left) cosets of H in G.(ii) Two right (resp. left) cosets of H in G are either disjoint or equal.(iii) For all a, b ∈ G,Ha = Hb⇔ ab−1 ∈ H and aH = bH ⇔ a−1b ∈ H .(iv) If R is the set of distinct right cosets of H in G and L is the set of distinct

left cosets of H in G, then |R| = |L|.

Definition 1.4.4. Let H be a subgroup of a group G. The index of H in G,denoted [G : H], is the cardinal number of the set of distinct right (resp. left)cosets of H in G.

Theorem 1.4.5. If K,H,G are groups with K < H < G, then[G : K] = [G : H][H : K]. If any two of these indices are finite, then so is thethird.

Corollary 1.4.6. (Lagrange). If H is a subgroup of a group G, then|G| = [G : H]|H|. In particular if G is finite, the order |a| of a ∈ G divides |G|.

If G is a group and H,K are subsets of G, we denote by HK the set{ab : a ∈ H, b ∈ K}; If H,K are subgroups, HK may not by a subgroup.

Theorem 1.4.7. Let H and K be finite subgroups of a group G. Then |HK| =|H||K|/|H ∩K|.

Proposition 1.4.8. If H and K are subgroups of a group G, then [H : H ∩K] ≤[G : K]. If [G : K] is finite, then [H : H∩K] = [G : K] if and only if G = KH .

4

CHAPTER 1. GROUPS

Proposition 1.4.9. Let H and K be subgroups of finite index of a group G. Then[G : H ∩ K] is finite and [G : H ∩ K] ≤ [G : H][G : K]. Furthermore,[G : H ∩K] = [G : H][G : K] if and only if G = HK.

1.5 NORMALITY, QUOTIENT GROUPS, AND

HOMOMORPHISMS

Theorem 1.5.1. If N is a subgroup of a group G, then the following conditionsare equivalent.

(i) Left and right congruence modulo N coincide (that is, define the sameequivalence relation on G);

(ii) every left coset of N in G is a right coset of N in G;(iii) aN = Na for all a ∈ G;(iv) for all a ∈ G, aNa−1 ⊂ N , where aNa−1 = {ana−1 : n ∈ N};(v) for all a ∈ G, aNa−1 = N .

Definition 1.5.2. A subgroup N of a group G which satisfies the equivalent con-ditions is said to be normal in G (or a normal subgroup of G); we write N CGif N is normal in G.

Every subgroup of an abelian group is trivially normal. The intersection of afamily of normal subgroups is a normal subgroup. If G is a group with subgroupsN and M such that N CM and M CG, it does not follow that N CG.

Theorem 1.5.3. Let K and N be subgroups of a group G with N normal in G.Then

(i) N ∩K is a normal subgroup of K;(ii) N is a normal subgroup of N ∨K;(iii) NK = N ∨K = KN ;(iv) if K is normal in G and K ∩ N = 〈e〉 then nk = kn for all k ∈ K and

n ∈ N .

Theorem 1.5.4. If N is a normal subgroup of a group G and G/N is the set ofall (left) cosets of N in G, then G/N is a group of order [G : N ] under the binaryoperation given by (aN)(bN) = abN .

If N is a normal subgroup of a group G, then the group G/N ,is called thequotient group or factor group of G by N. If written additively, we have(a+N) + (b+N) = (a+ b) +N .

Theorem 1.5.5. If f : G→ H is a homomorphism of groups, then the kernel off is a normal subgroup of G. Conversely, if N is a normal subgroup of G, thenthe map π : G→ G/N given by π(a) = aN is an epimorphism with kernel N.

The map π : G→ G/N is called the canonical epimorphism or projection

5

CHAPTER 1. GROUPS

Theorem 1.5.6. If f : G→ H is a homomorphism of groups and N is a normalsubgroup of G contained in the kernel of f, then there is a unique homomorphismf : G/N → H such that f(aN) = f(a) for all a ∈ G. Imf = Imf andKerf = (Kerf)/N . f is an isomorphism if and only if f is an epimorphism andN = Kerf .

Theorem 1.5.7. (First Isomorphism Theorem) If f : G→ H is a homomorphismof groups, then f induces an isomorphism G/Kerf ∼= Imf .

Corollary 1.5.8. If f : G→ H is a homomorphism of groups, N CG,M CH ,and f(N) < M , then f induces a homomorphism f : G/N → H/M , givenby aN 7→ f(a)M.f is an isomorphism if and only if Imf ∨ M = H andf−1(M) ⊂ N . In particular if f is an epimorphism such that f(N) = M andKerf ⊂ N , then f is an isomorphism.

Corollary 1.5.9. (Second Isomorphism Theorem) If K and N are subgroups of agroup G, with N normal in G, then K/(N ∩K) ∼= NK/N .

Corollary 1.5.10. (Third Isomorphism Theorem) If H and K are normal sub-groups of a group G such that K < H , then H/K is a normal subgroup of G/Kand (G/K)/(H/K) ∼= G/H .

Theorem 1.5.11. If f : G → H is an epimorphism of groups, then the assign-ment K 7→ f(K) defines a one-to-one correspondence between the set Sf (G) ofall subgroups K of G which contain Ker f and the set S(H) of all subgroups ofH. Normal subgroups correspond to normal subgroups.

Corollary 1.5.12. If N is a normal subgroup of a group G, then every subgroupG/N is of the form K/N where K is a subgroup of G that contains N. Further-more, K/N is normal in G/N if and only if K is normal in G.

1.6 SYMMETRIC, ALTERNATING, AND DIHEDRAL GROUPS

The symmetric group Sn is the group of all bijections In → In, where In ={1, 2, . . . , n}. The elements of Sn are called permutations.

Definition 1.6.1. Let i1, i2, . . . , ir, (r ≤ n) be distinct elements of In = {1, 2, . . . , n}.Then (i1i2i3 . . . ir) denotes the permutation that maps i1 7→ i2, i2 7→ i3, . . . , ir 7→i1, and maps every other element of In onto itself. (i1i2 · · · ir) is called a cycleof length r or an r-cycle; a 2-cycle is called a transposition.

A 1-cycle (k) is the identity permutation. Clearly, an r-cycle is an element oforder r in Sn. The inverse of the cycle (i1i2 · · · ir) is the cycle (irir−1ir−2 · · · i2i1) =(i1irir−1 · · · i2).

6

CHAPTER 1. GROUPS

Definition 1.6.2. The permutations σ1, σ2, . . . , σrofSn are said to be disjointprovided that for each 1 ≤ i ≤ r, and every k ∈ In, σi(k) 6= k implies σj(k) = kfor all j 6= i.

In other words σ1, σ2, . . . , σr are disjoint if and only if no element of In ismoved by more than one of them. στ = τσ whenever σ and τ are disjoint.

Theorem 1.6.3. Every nonidentity permutation in Sn is uniquely (up to the orderof the factors) a product of disjoint cycles, each of which has length at least 2.

Corollary 1.6.4. The order of a permutation σ ∈ Sn is the least common multi-ple of the orders of its disjoint cycles.

Corollary 1.6.5. Every permutation in Sn can be written as a product of (notnecessarily disjoint) transpositions.

Definition 1.6.6. A permutation τ ∈ Sn is said to be even (resp. odd) if τ canbe written as a product of an even (resp. odd) number of transpositons.

Theorem 1.6.7. A permutation in Sn(n ≥ 2) cannot be both even and odd.

Theorem 1.6.8. For each n ≥ 2, let An be the set of all even permutations ofSn. Then An is a normal subgroup of Sn of index 2 and order |Sn|/2 = n!/2.Furthermore An is the only subgroup of Sn of index 2.

The group An is called the alternating group on n letters

Definition 1.6.9. A group G is said to be simple if G has no proper normalsubgroups.

Theorem 1.6.10. The alternating group An is simple if and only if n 6= 4.

Recall that if τ is a 2-cycle, τ 2 = (1) and hence τ = τ−1.

Dn is called the dihedral group of degree n. It is isomorphic to and usuallyidentified with the group of all symmetries of a regular polygon with n sides.

Theorem 1.6.11. For each n ≥ 3 the dihedral group Dn is a group of order 2nwhose generators a and b satisfy:

(i) an = (1), b2 = (1), ak 6= (1) if 0 < k < n;(ii) ba = a−1b.

1.7 DIRECT PRODUCTS AND DIRECT SUMS

Definition 1.7.1. Let G1, G2 . . . , Gn be a finite collection of groups. The exter-nal direct product ofG1, G2, . . . , Gn, written asG1⊕G2⊕· · ·⊕Gn, is the set ofall n-tuples for which the ith component is an element of Gi and the operation iscomponentwise. In symbols, G1⊕G2⊕ · · · ⊕Gn = {(g1, g2, . . . , gn) : gi ∈ Gi}

7

CHAPTER 1. GROUPS

Theorem 1.7.2. If {Gi : i ∈ I} is a family of groups, then(i) the direct product

∏i∈I Gi is a group;

(ii) for each k ∈ I the map πk :∏

i∈I Gi → Gk given by f 7→ f(k) (or{ai} 7→ ak) is an epimorphism of groups. This is often called the canonicalprojection of the direct product.

Proposition 1.7.3. If G1, . . . , Gn are groups, their direct product is a group oforder |G1||G2| · · · |Gn| (if any Gi is infinite, so is the direct product.

Proposition 1.7.4. Let G1, G2, . . . , Gn be groups and let G = G1× · · ·×Gn betheir direct product.

(i) For each fixed i the set of elements of G which have the identity ofGj in thejth position for all j 6= i and arbitrary elements of Gi in position i is a subgroupof G isomorphic toGi: Gi

∼= {(1, 1, . . . , 1, gi, 1, . . . , 1) : gi ∈ Gi}. If we identifyGi with this subgoup, thenGiCG andG/Gi

∼= G1×· · ·×Gi−1×Gi+1×· · ·×Gn.(ii) For each fixed i define πi : G → Gi by πi((g1, g2, . . . , gn)) = gi. Then

πi is a surjective homomorphism with kerπi = {(g1, . . . , gi−1, 1, g1+1, . . . , gn) :gj ∈ Gj for all j 6= i} ∼= G1×· · ·×Gi−1×Gi+1×· · ·×Gn (here the 1 appearsin position i).

(iii) Under the identification in part (i), if x ∈ Gi and y ∈ Gj for some i 6= j,then xy = yx.

Corollary 1.7.5. An external direct productG1×G2×· · ·×Gn of a finite numberof finite cyclic groups is cyclic if and only if |Gi| and |Gj| are relatively primewhen i 6= j.

Definition 1.7.6. Let H1, H2, . . . , Hn be a finite collection of normal subgroupsof G. We say that G is the internal direct product of H1, H2, . . . , Hn and writeG = H1 ×H2 × · · · ×Hn, if

(i) G = H1H2 · · ·Hn = {h1h2 · · ·hn : hi ∈ Hi}(ii) (H1H2 · · ·Hi) ∩Hi+1 = {e} for i = 1, 2, . . . , n− 1.

Theorem 1.7.7. If a group G is the internal direct product of a finite number ofsubgroups H1, H2, . . . , Hn, then G is isomorphic to the external direct productof H1, H2, . . . , Hn. So H1 ×H2 × · · · ×Hn

∼= H1 ⊕H2 ⊕ · · · ⊕Hn.

Theorem 1.7.8. For {Gi : i ∈ I} a family of groups, define∏w

i∈I Gi (calledthe weak direct product to be the set of f ∈

∏i∈I Gi such that f(i) = ei, the

identity in Gi, for all but a finite number of i, then(i)∏w

i∈I Gi is a normal subgroup of∏

i∈I Gi;(ii) for each k ∈ I the map ik : Gk →

∏wi∈I Gi given by ik(a) = {ai}i∈I ,

where ai = e for i 6= k and ak = a, is a monomorphism of groups;(iii) for each i ∈ I, ii(Gi) is a normal subgroup of

∏i∈I Gi.

The maps ik are called the canonical injections.

Theorem 1.7.9. Let {Ni : i ∈ I} be a family of normal subgroups of a group Gsuch that

8

CHAPTER 1. GROUPS

(i) G = 〈⋃i∈I Ni〉;

(ii) for each k ∈ I,Nk ∩ 〈⋃i 6=kNi〉 = 〈e〉.

ThenG ∼=∏w

i∈I Ni. Here 〈N1∪N2∪· · ·∪Nr〉 = N1N2 · · ·Nr = {n1n2 · · ·nr :ni ∈ Ni}.

Corollary 1.7.10. If N1, N2, . . . , Nr are normal subgroups of a group G suchthatG = N1N2 · · ·Nr and for each 1 ≤ k ≤ r,Nk∩(N1 · · ·Nk−1Nk+1 · · ·Nr) =〈e〉, then G ∼= N1 ×N2 × · · · ×Nr.

Proposition 1.7.11. Let {Gi : i ∈ I} and {Ni : i ∈ I} be families of groupssuch that Ni is a normal subgroup of Gi for each i ∈ I .

(i)∏

i∈I Ni is a normal subgroup of∏

i∈I Gi and∏

i∈I Gi /∏

i∈I Ni∼=∏

i∈I Gi/Ni.(ii)

∏wi∈I Ni is a normal subgroup of

∏wi∈I Gi and

∏wi∈I Gi /

∏wi∈I Ni

∼=∏wi∈I Gi/NI .

9

CHAPTER 1. GROUPS

10

Chapter 2

The Structure of Groups

2.1 FREE ABELIAN GROUPS

A basis of an abelian group F is a subset X of F such that (i) F = 〈X〉; and (ii) fordistinct x1, x2, . . . , xk ∈ X and ni ∈ Z, n1x1+n2x2+ · · ·+nkxk = 0⇒ ni = 0for every i.

An abelian group F that satisfies the following condidtions is called a freeabelian group (on the set X)

Theorem 2.1.1. The following conditions on an abelian group F are equivalent.(i) F has a nonempty basis.(ii) F is the internal direct sum of a family of infinite cyclic subgroups.(iii) F is isomorphic to a direct sum of copies of the additive group Z.(iv) There exists a nonempty set X and a function i : X → F with the follow-

ing property: given an abelian group G and function f : X → G, there exists aunique homomorphism of groups f : F → G such that fi = f .

Theorem 2.1.2. Any two bases of a free abelian group F have the same cardi-nality.

The cardinal number of a basis X, |X|, is called the rank of F.

Proposition 2.1.3. Let F1 be the free abelian group on the set X1 and F2 thefree abelian group on the set X2. Then F1

∼= F2 if and only if F1 and F2 havethe same rank.

Theorem 2.1.4. Every abelian group G is the homomorphic image of a freeabelian group of rank |X|, where X is a set of generators of G.

Theorem 2.1.5. If F is a free abelian group of finite rank n and G is a nonzerosubgroup of F, then there exists a basis {x1, . . . , xn} of F, an integer r (1≤ r ≤ n)and positive integers d1, . . . , dr such that d1|d2| · · · |dr and G is free abelian withbasis {d1x1, . . . , drxr}.

Corollary 2.1.6. If G is a finitely generated abelian group generated by n ele-ments, then every subgroup H of G may be generated by m elements withm ≤ n.

11

CHAPTER 2. THE STRUCTURE OF GROUPS

2.2 FINITELY GENERATED ABELIAN GROUPS

Theorem 2.2.1. Every finitely generated abelian group G is isomorphic to afinite direct sum of cyclic groups in which the finite cyclic summands (if any) areof orders m1, . . . ,mt, where m1 > 1 and m1|m2| · · · |mt.

Theorem 2.2.2. Every finitely generated abelian group G is isomorphic to afinite direct sum of cyclic groups, each of which is either infinite or of order apower of a prime.

Lemma 2.2.3. If m is a positive integer and m = pn11 p

n22 · · · pnt

t (p1, . . . , pt dis-tinct primes and each ni > 0), then Zm ∼= Zpn1

1⊕ Zpn2

2⊕ · · · ⊕ Zpnt

t.

We have that Zrn ∼= Zr ⊕ Zn whenever (r, n) = 1.

Corollary 2.2.4. If G is a finite abelian group of order n, then G has a subgroupof order m for every positive integer m that divides n.

Lemma 2.2.5. Let G be an abelian group, m an integer and p a prime integer.Then each of the following is a subgroup of G:

(i) mG = {mu : u ∈ G};(ii) G[m] = {u ∈ G : mu = 0};(iii) G(p) = {u ∈ G : |u| = pn for some n ≥ 0};(iv) Gt = {u ∈ G : |u| is finite }.In particular there are isomorphisms(v) Zpn [p] ∼= Zp(n ≥ 1) and pmZpn ∼= Zpn−m(m < n).Let H and Gi(i ∈ I) be abelian groups.(vi) If g : G →

∑i∈I Gi is an isomorphism, then the restriction of g to

mG and G[m] respectively are isomorphisms mG ∼=∑

i∈I mGi and G[m] ∼=∑i∈I Gi[m].(vii) If f : G → H is an isomorphism, then the restrictions of f to Gt and

G(p) respectively are isomorphisms Gt∼= Ht and G(p) ∼= H(p).

If G is an abelian group, then the subgroup Gt is called the torsion subgroupof G. If G = Gt, then G is said to be a torsion group. If Gt = 0 then G is saidto be torsion-free.

Theorem 2.2.6. Let G be a finitely generated abelian group.(i) There is a unique nonnegative integer s such that the number of infinite

cyclic summands in any decomposition of G as a direct sum of cyclic groups isprecisely s;

(ii) either G is free abelian or there is a unique list of (not necessarily distinct)positive integers m1, . . . ,mt such that m1 > 1,m1|m2| · · · |mt and G ∼= Zm1 ⊕· · · ⊕ Zmt ⊕ F with F free abelian;

12


(iii) either G is free abelian or there is a list of positive integers ps11 , . . . , pskk ,

which is unique except for the order of its members, such that p1, . . . , pk are(not necessarily distinct) primes, s1, . . . , sk are (not necessarily distinct) positiveintegers and G ∼= Zps11 ⊕ . . .⊕ Zpskk

⊕ F with F free abelian.

If G is a finitely generated abelian group, then the uniquely determined inte-gers m1, . . . ,mt as in previous theorem, are called the invariant factors of G.The uniquely determined prime powers are called the elementary divisors of G.

Corollary 2.2.7. Two finitely generated abelian groups G and H are isomorphicif and only if G/Gt and H/Ht have the same rank and G and H have the sameinvariant factors (resp. elementary divisors).

2.3 THE KRULL-SCHMIDT THEOREM

Every finitely generated abelian group is the direct sum of a finite number ofindecomposable groups and these indecomposable summands are uniquely de-termined up to isomorphism.

Definition 2.3.1. A group G is indecomposable if G 6= 〈e〉 and G is not theinternal direct product of two of its proper subgroups.

Thus G is indecomposable if and only if G 6= 〈e〉 and G ∼= H × K impliesH = 〈e〉 or K = 〈e〉. For example, every simple group is indecomposable. AlsoZ,Zpn , and Sn are indecomposable but not simple.

Definition 2.3.2. A group G is said to satisfy the ascending chain condition(ACC) on (normal) subgroups if for every chain G1 < G2 < · · · of (normal)subgroups of G there is an integer n such that Gi = Gn for all i ≥ n. G is saidto satisfy the descending chain condition (DCC) on (normal) subgroups if forevery chain G1 > G2 > · · · of (normal) subgroups of G there is an integer nsuch that Gi = Gn for all i ≥ n.

Every finite group satisfies both chain conditions

Theorem 2.3.3. If a group G satisfies either the ascending or descending chaincondition on normal subgroups, then G is the direct product of a finite numberof indecomposable subgroups.

An endomorphism f of a group G is called a normal endomorphism ifaf(b)a−1 = f(aba−1) for all a, b ∈ G.

Lemma 2.3.4. Let G be a group that satisfies the ascending (resp. descending)chain condition on normal subgroups and f a (normal) endomorphism of G. Thenf is an automorphism if and only if f is an epimorphism (resp. monomorphism).

13


Lemma 2.3.5. (Fitting) If G is a group that satisfies both the ascending and de-scending chain conditions on normal subgroups and f is a normal endomorphismof G, then for some n ≥ 1, G = Kerfn × Imfn.

An endomorphism f of a group G is said to be nilpotent if there exists apositive integer n such that fn(g) = e for all g ∈ G.

Corollary 2.3.6. If G is an indecomposable group that satisfies both the ascend-ing and descending chain conditions on normal subgroups and f is a normalendomorphism of G, then either f is nilpotent or f is an automorphism.

Corollary 2.3.7. Let G ( 6= 〈e〉) be an indecomposable group that satisfies boththe ascending and descending chain conditions on normal subgroups. If f1, . . . , fnare normal nilpotent endomorphisms of G such that every fi1 + · · · + fir(1 ≤i1 < i2 < · · · < ir ≤ n) is an endomorphism, then f1+f2+ · · ·+fn is nilpotent.

Theorem 2.3.8. (Krull-Schmidt) Let G be a group that satisfies both (ACC) and(DCC) on normal subgroups. If G = G1 ×G2 × · · · ×Gs and G = H1 ×H2 ×· · · × Ht with each Gi, Hj indecomposable, then s = t and after reindexingGi∼= Hi for every i and for each r < t. G = G1× · · · ×Gr ×Hr+1× · · · ×Ht.

2.4 THE ACTION OF A GROUP ON A SET

Definition 2.4.1. An action of a group G on a set S is a function G × S → S(usually denoted by (g, x) 7→ gx) such that for all x ∈ S and g1, g2 ∈ G:ex = x and (g1g2)x = g1(g2x)When such an action is given, we say that G acts on the set S. The notation

gx is ambiguous.

As an example, let H be a subgroup of a group G. An action of H on the setG is given by (h, x) 7→ hxh−1; to avoid confusion with the product in G, thisaction of h ∈ H is always denoted hxh−1 and not hx. This action of h ∈ H onG is called conjugation by h and the element hxh−1 is said to be a conjugate ofx.

Theorem 2.4.2. Let G be a group that acts on a set S.(i) The relation on S defined by x ∼ x′ ⇔ gx = x′ for some g ∈ G is an

equivalence relation.(ii) For each x ∈ S, Gx = {g ∈ G : gx = x} is a subgroup of G.

The equivalence classes are called the orbits of G on S; the orbit of x ∈ Sis denoted x. The subgroup Gx is called variously the subgroup fixing x, theisotropy group of x or the stabilizer of x.

For some examples, if a group G acts on itself by conjugation, then the orbit{gxg−1 : g ∈ G} of x ∈ G is called the conjugacy class of x. If a subgroup

14


H acts on G by conjugation the stabilizer group Hx = {h ∈ H : hxh−1 =x} = {h ∈ H : hx = xh} is called the centralizer of x in H and is denotedCH(x). If H = G,CG(x) is simply called the centralizer of x. If H acts byconjugation on the set S of all subgroups of G, then the subgroup of H fixingK ∈ S namely {h ∈ H : hKh−1 = K} is called the normalizer of K in Hand denoted NH(K). The group NG(K) is simply called the normalizer of K.Clearly every subgroup K is normal in NG(K); K is normal in G if and only ifNG(K) = G.

Theorem 2.4.3. If a group G acts on a set S, then the cardinal number of theorbit of x ∈ S is the index [G : Gx].

Corollary 2.4.4. Let G be a finite group and K a subgroup of G.(i) The number of elements in the conjugacy class x ∈ G is [G : CG(x)],

which divides |G|;(ii) if x1, . . . , xn(xi ∈ G) are the distinct conjugacy cliasses of G, then

|G| =n∑i=1

[G : CG(xi)];

.This equation is know as the class equation of the finite group G.

(iii) the number of subgroups of G conjugate to K is [G : NG(K)], whichdivides |G|.

Theorem 2.4.5. If a group G acts on a set S, then this action induces a homo-morphism G→ A(S), where A(S) is the group of all permutations of S.

Corollary 2.4.6. (Cayley) If G is a group, then there is a monomorphism G →A(G). Hence every group is isomorphic to a group of permutations. In particu-lar every finite group is isomorphic to a subgroup of Sn with n = |G|.

Corollary 2.4.7. Let G be a group.(i) For each g ∈ G, conjugation by g induces an automorphism of G.(ii) There is a homomorphism G→ AutG whose kernel is Z(G) = {g ∈ G :

gx = xg for all x ∈ G}.

If G acts on itself by conjugation then for each g ∈ G, the map φg : G → Ggiven by φg(x) = gxg−1 is an automorphism, called the inner automorphisminduced by g. The normal subgroup Z(G) = Kerφ is called the center of G.

Proposition 2.4.8. Let H be a subgroup of a group G and let G act on the setS of all left cosets of H in G by left translation. Then the kernel of the inducedhomomorphism G→ A(S) is contained in H.

Corollary 2.4.9. If H is a subgroup of index n in a group G and no nontrivialnormal subgroup of G is contained in H, then G is isomorphic to a subgroup ofSn.

15


Corollary 2.4.10. If H is a subgroup of a finite group G of index p, where p isthe smallest prime dividing the order of G, then H is normal in G.

2.5 THE SYLOW THEOREMS

Lemma 2.5.1. If a group H of order pn (p prime) acts on a finite set S and ifS0 = {x ∈ S : hx = x for all h ∈ H}, then |S| ≡ |S0| (mod p).

Theorem 2.5.2. (Cauchy) If G is a finite group whose order is divisible by aprime p, then G contains an element of order p.

A group in which every element has order a power (≥ 0) of some fixed primep is called a p-group. If H is a subgroup of a group G and H is a p-group, H issaid to be a p-subgroup of G.

Corollary 2.5.3. A finite group G is a p-group if and only if |G| is a power of p.

Corollary 2.5.4. The center Z(G) of a nontrivial finite p-group G contains morethan one element.

Lemma 2.5.5. If H is a p-subgroup of a finite group G, then [NG(H) : H] ≡[G : H](mod p).

Corollary 2.5.6. If H is a p-subgroup of a finite group G such that p divides[G : H], then NG(H) 6= H .

Theorem 2.5.7. (First Sylow Theorem) Let G be a group of order pnm, withn ≥ 1, p prime, and (p,m) = 1. Then G contains a subgroup of order pi foreach 1 ≤ i ≤ n and every subgroup of G of order pi (i < n) is normal in somesubgroup of order pi+1.

A subgroup P of a group G is said to be a Sylow p-subgroup (p prime) if Pis a maximal p-subgroup of G (that is, P < H < G with H a p-group impliesP = H). Sylow p-subgroups always exist, though they may be trivial, and everyp-subgroup is contained in a Sylow p-subgroup. A finite group G has a nontrivialSylow p-subgroup for every prime p that divides |G|.

Corollary 2.5.8. Let G be a group of order pnm with p prime, n ≥ 1 and(m, p) = 1. Let H be a p-subgroup of G.

(i) H is a Sylow p-subgroup of G if and only if |H| = pn.(ii) Every conjugate of a Sylow p-subgroup is a Sylow p-subgroup.(iii) If there is only one Sylow p-subgroup P, then P is normal in G.

Theorem 2.5.9. (Second Sylow Theorem) If H is a p-subgroup of a finite groupG, and P is any Sylow p-subgroup of G, then there exists x ∈ G such thatH < xPx−1. In particular, any two Sylow p-subgroups of G are conjugate.

16


Theorem 2.5.10. (Third Sylow Theorem) If G is a finite group and p a prime,then the number of Sylow p-subgroups of G divides |G| and is of the form kp+ 1for some k ≥ 0.

Theorem 2.5.11. If P is a Sylow p-subgroup of a finite group G, thenNG(NG(P )) =NG(P ).

2.6 CLASSIFICATION OF FINITE GROUPS

Proposition 2.6.1. Let p and q be primes such that p > q. If q - p − 1, thenevery group of order pq is isomorphic to the cyclic group Zpq. If q|p − 1, thenthere are (up to isomorphism) exactly two distinct groups of order pq: the cyclicgroup Zpq and a nonabelian group K generated by elements c and d such that

|c| = p; |d| = q; dc = csd,

where s 6≡ 1 (mod p) and sq ≡ 1 (mod p).

Corollary 2.6.2. If p is an odd prime, then every group of order 2p is isomorphiceither to the cyclic group Z2p or the dihedral group Dp.

2.7 NILPOTENT AND SOLVABLE GROUPS

Definition 2.7.1. (i) For any (finite or infinite) group G define the following sub-groups inductively: Z0(G) = 1, Z1(G) = Z(G) and Zi+1(G) is the subgroupof G containing Zi(G) such that Zi+1(G) / Zi(G) = Z(G/Zi(G)) (i.e., Zi+1(G)is the complete preimage in G of the center of G/Zi(G) under the natural pro-jection). The chain of subgroups Z0(G) < Z1(G) < Z2(G) < · · · is called theupper central series of G. (ii) A group G is called nilpotent if Zc(G) = G forsome c ∈ Z. The smallest such c is called the nilpotence class of G.

Proposition 2.7.2. Let p be a prime and let P be a group of order pa. Then P isnilpotent of nilpotence class at most a− 1.

Theorem 2.7.3. Let G be a finite group, let p1, p2, . . . , ps be the distinct primesdividing its order and let Pi ∈ Sylpi(G), 1 ≤ i ≤ s. Then the following areequivalent:

(i) G is nilpotent.(ii) If H < G then H < NG(H), i.e., every proper subgroup of G is a proper

subgroup of its normalizer in G.(iii) Pi CG for 1 ≤ i ≤ s, i.e., every Sylow subgroup is normal in G.(iv) G ∼= P1 × P2 × · · · × Ps.

17


Corollary 2.7.4. A finite abelian group is the direct product of its Sylow sub-groups.

Proposition 2.7.5. If G is a finite group such that for all positive integers ndividing its order, G contains at most n elements x satisfying xn = 1, then G iscyclic.

Proposition 2.7.6. (Frattini’s Argument) Let G be a finite group, let H be a nor-mal subgroup of G and let P be a Sylow p-subgroup of H. Then G = HNG(P )and [G : H] divides [NG(P )].

Proposition 2.7.7. A finite group is nilpotent if and only if every maximal sub-group is normal.

Corollary 2.7.8. If G is a finite nilpotent group and m divides |G|, then G has asubgroup of order m.

Theorem 2.7.9. The direct product of a finite number of nilpotent groups isnilpotent.

Definition 2.7.10. Let G be a group. The subgroup of G generated by the set{aba−1b−1 : a, b ∈ G} is called the commutator subgroup of G and denotedG’.

Theorem 2.7.11. If G is a group, then G’ is a normal subgroup of G and G/G′

is abelian. If N is a normal subgroup of G, then G/N is abelian if and only if Ncontains G’.

Definition 2.7.12. For any (finite or infinite) group G define the following sub-groups inductively: G(0) = G, G(1) = G′ and G(i) = (G(i−1))′. G(i) is calledthe ith derived subgroup of G. The chain of groups G(0) ≥ G(1) ≥ G(2) ≥ · · ·is called the lower central series of G. (The term “lower” indicates that G(i) ≥G(i+1). Each G(i) is a normal subgroup of G.

Definition 2.7.13. A group G is said to be solvable if G(n) = 〈e〉 for some n.

Proposition 2.7.14. Every nilpotent group is solvable.

Theorem 2.7.15. (i) Every subgroup and every homomorphic image of a solv-able group is solvable. (ii) If N is a normal subgroup of a group G such that Nand G/N are solvable, then G is solvable.

Corollary 2.7.16. If n ≥ 5, then the symmetric group Sn is not solvable.

Proposition 2.7.17. (P. Hall) Let G be a finite solvable group of order mn, with(m,n) = 1. Then

(i) G contains a subgroup of order m;(ii) any two subgroups of G of order m are conjugate;(iii) any subgroup of G of order k, where k|m, is contained in a subgroup of

order m.

18


2.8 NORMAL AND SUBNORMAL SERIES

Definition 2.8.1. A subnormal series of a group G is a chain of subgroupsG = G0 > G1 > · · · > Gn such that Gi+1 is normal in Gi for 0 ≤ i < n. Thefactors of the series are the quotient groups Gi/Gi+1. The length of the seriesis the number of strict inclusions (or alternatively, the number of nonidentityfactors). A subnormal series such that Gi is normal in G for all i, is said to be anormal series.

Definition 2.8.2. Let G = G0 > G1 > · · · > Gn be a subnormal series. Aone-step refinement of this series is any series of the form G = G0 > · · · >Gi > N > Gi+1 > · · · > Gn or G = G0 > · · · > Gn > N , where N is a normalsubgroup of Gi and (if i < n) Gi+1 is normal in N. A refinement of a subnormalseries S is any subnormal series obtained from S by a finite sequence of one-steprefinements. A refinement of S is said to be proper if its length is larger than thelength of S.

Definition 2.8.3. A subnormal series G = G0 > G1 > · · · > Gn = 〈e〉is a composition series if each factor Gi/Gi+1 is simple. A subnormal seriesG = G0 > G1 > · · · > Gn = 〈e〉 is a solvable series if each factor is abelian.

Theorem 2.8.4. (i) Every finite group G has a composition series. (ii) Everyrefinement of a solvable series is a solvable series. (iii) A subnormal series is acomposition series if and only if it has no proper refinements.

Theorem 2.8.5. A group G is solvable if and only if it has a solvable series.

Proposition 2.8.6. A finite group G is solvable if and only if G has a compositionseries whose factors are cyclic of prime order.

Definition 2.8.7. Two subnormal series S and T of a group G are equivalent ifthere is a one-to-one correspondence between the nontrivial factors of S and thenontrivial factors of T such that corresponding factors are isomorphic groups.

Lemma 2.8.8. If S is a composition series of a group G, then any refinement ofS is equivalent to S.

Lemma 2.8.9. (Zassenhaus) Let A∗, A,B∗, B be subgroups of a group G suchthat A∗ is normal in A and B∗ is normal in B.

(i) A∗(A ∩B∗) is a normal subgroup of A∗(A ∩B);(ii) B∗(A∗ ∩B) is a normal subgroup of B∗(A ∩B);(iii) A∗(A ∩B)/A∗(A ∩B∗) ∼= B∗(A ∩B)/B∗(A∗ ∩B).

Theorem 2.8.10. (Schreier) Any two subnormal (resp. normal) series of a groupG have subnormal (resp. normal) refinements that are equivalent.

Theorem 2.8.11. (Jordan-Holder) Any two composition series of a group G areequivalent. Therefore every group having a composition series determines aunique list of simple groups.

19


20

Chapter 3

Rings

3.1 RINGS AND HOMOMORPHISMS

Definition 3.1.1. A ring is a nonempty set R together with two binary operations(usually denoted as addition (+) and multiplication) such that:

(i) (R,+) is an abelian group;(ii) (ab)c = a(bc) for all a, b, c ∈ R (associative multiplication);(iii) a(b + c) = ab + ac and (a + b)c = ac + bc (left and right distributive

laws).If in addition:

(iv) ab = ba for all a, b ∈ Rthen R is said to be a commutative ring. If R contains an element 1R such that

(v) 1Ra = a1R = a for all a ∈ R,then R is said to be a ring with identity.

The additive identity element of a ring is called the zero element and denoted0.

Theorem 3.1.2. Let R be a ring. Then(i) 0a = a0 = 0 for all a ∈ R;(ii) (−a)b = a(−b) = −(ab) for all a, b ∈ R;(iii) (−a)(−b) = ab for all a, b ∈ R;(iv) (na)b = a(nb) = n(ab) for all n ∈ Z and all a, b ∈ R;(v) (

∑ni=1 ai)(

∑mj=1 bj) =

∑ni=1

∑mj=1 aibj for all ai, bj ∈ R.

Definition 3.1.3. A nonzero element a in a ring R is said to be a left (resp. right)zero divisor if there exists a nonzero b ∈ R such that ab = 0 (resp. ba = 0). Azero divisor is an element of R which is both a left and a right zero divisor.

A ring R has no zero divisors if and only if the right and left cancellation lawshold in R; that is, for all a, b, c ∈ R with a 6= 0, ab = ac or ba = ca⇒ b = c.

Definition 3.1.4. An element a in a ring R with identity is said to be left (resp.right) invertible if there exists c ∈ R (resp. b ∈ R) such that ca = 1R (resp.

CHAPTER 3. RINGS

ab = 1R). The element c (resp. b) is called a left (resp. right) inverse of a. Anelement a ∈ R that is both left and right invertible is said to be invertible or tobe a unit.

(i) The left and right inverses of a unit a in a ring R with identity necessarilycoincide. (ii) The set of units in a ring R with identity forms a group undermultiplication.

Definition 3.1.5. A commutative ring R with identity 1R 6= 0 and no zero divisorsis called an integral domain. A ring D with identity 1D 6= 0 in which everynonzero element is a unit is called a division ring. A field is a commutativedivision ring.

(i) Every integral domain and every division ring has at least two elements(namely 0 and 1R). (ii) A ring R with identity is a division ring if and only if thenonzero elements of R form a group under multiplication. (iii) Every field F isan integral domain.

Theorem 3.1.6. (Binomial Theorem) Let R be a ring with identity, n a positiveinteger, and a, b, a1, a2, . . . , as ∈ R.

(i) If ab = ba, then (a+ b)n =∑n

k=0

(nk

)akbn−k;

(ii) If aiaj = ajai for all i and j, then

(a1 + a2 + · · ·+ as)n =

∑ n!

(i1!) · · · (in!)ai11 a

i22 · · · aiss

where the sum is over all s-tuples (i1, i2, . . . , is) such that i1 + i2 + · · ·+ is = n.

Definition 3.1.7. Let R and S be rings. A function f : R → S is a homomor-phism of rings provided that for all a, b ∈ R:

f(a+ b) = f(a) + f(b) and f(ab) = f(a)f(b)

.

A homomorphism of rings is, in particular, a homomorphism of the under-lying additive groups. The same terminology is used: monomorphism, epimor-phism, isomorphism, etc. A monomorphism of ringsR→ S is sometimes calledan embedding of R in S. The kernel of a homomorphism of rings f : R → Sis its kernel as a map of additive groups; that is ker f = {r ∈ R : f(r) = 0}.Similarly the image of f, denoted Im f, is {s ∈ S : s = f(r) for some r ∈ R}. IfR and S both have identities 1R and 1S , we do not require that a homomorphismof rings map 1R to 1S .

Definition 3.1.8. Let R be a ring. If there is a least positive integer n such thatna = 0 for all a ∈ R, then R is said to have characteristic n. If no such n existsR is said to have characteristic zero. (Notation: char R = n).

22

CHAPTER 3. RINGS

Theorem 3.1.9. Let R be a ring with identity 1R and characteristic n > 0.(i) If φ : Z → R is the map given by m 7→ m1R, then φ is a homomorphism

of rings with kernel 〈n〉 = {kn : k ∈ Z}.(ii) n is the least positive integer such that n1R = 0.(iii) If R has no zero divisors (in particular if R is an integral domain), then n

is prime.

Theorem 3.1.10. Every ring R may be embedded in a ring S with identity. Thering S (which is not unique) may be chosen to be either of characteristic zero orof the same characteristic as R.

3.2 IDEALS

Definition 3.2.1. Let R be a ring and S a nonempty subset of R that is closedunder the operations of addition and multiplication in R. If S is itself a ringunder these operations then S is called a subring of R. A subring I of a ring R isa left ideal provided

r ∈ R and x ∈ I ⇒ rx ∈ I;

I is a right ideal provided

r ∈ R and x ∈ I ⇒ xr ∈ I;

I is an ideal if it is both a left and right ideal.

Two ideals of a ring R are R itself and the trivial ideal (denoted 0). A (left)ideal I of R such that I 6= 0 and I 6= R is called a proper (left) ideal. Observethat if R has an identity 1R and I is a (left) ideal of R, then I = R if and onlyif 1R ∈ I . Consequently, a nonzero (left) ideal I of R is proper if and only if Icontains no units of R; (for if u ∈ R is a unit and u ∈ I , then 1R = u−1u ∈ I).In particular, a division ring D has no proper left (or right) ideals since everynonzero element of D is a unit.

Theorem 3.2.2. A nonempty subset I of a ring R is a left (resp. right) ideal ifand only if for all a, b ∈ I and r ∈ R:

(i) a, b ∈ I ⇒ a− b ∈ I; and(ii) a ∈ I, r ∈ R⇒ ra ∈ I (resp. ar ∈ I).

Corollary 3.2.3. Let {Ai : i ∈ I} be a family of (left) ideals in a ring R. Then⋂i∈I Ai is also a (left) ideal

Definition 3.2.4. Let X be a subset of a ring R. Let {Ai : i ∈ I} be the familyof all (left) ideals in R which contain X. Then

⋂i∈I Ai is called the (left) ideal

generated by X. This is denoted (X).

23

CHAPTER 3. RINGS

The elements of X are called generators of the ideal (X). IfX = {x1, . . . , xn},then the ideal (X) is denoted (x1, x2, . . . , xn) and said to be finitely generated.An ideal (x) generated by a single element is called a principal ideal. A prin-ciple ideal ring is a ring in which every ideal is principle. A principle ideal ringwhich is an integral domain is called a principle ideal domain.

Theorem 3.2.5. Let R be a ring a ∈ R and X ⊂ R.(i) The principle ideal (a) consists of all elements of the form ra+ as+na+∑mi=1 riasi (r, s, ri, si ∈ R;m ∈ N∗; and n ∈ Z).(ii) If R has an identity, then (a) = {

∑ni=1 riasi : ri, si ∈ R;n ∈ N∗}.

(iii) If a is in the center of R, then (a) = {ra+ na : r ∈ R, n ∈ Z}.(iv) Ra = {ra : r ∈ R} (resp. aR = {ar : r ∈ R}) is a left (resp. right)

ideal in R (which may not contain a). If R has an identity, then a ∈ Ra anda ∈ aR.

(v) If R has an identity and a is in the center of R, then Ra = (a) = aR.(vi) If R has an identity and X is in the center of R, then the ideal (X) consists

of all finite sums r1a1 + · · ·+ rnan (n ∈ N∗; ri ∈ R; ai ∈ X).

Let A1, A2, . . . , An be nonempty subsets of a ring R. Denote by A1 + A2 +· · ·+An the set {a1 +a2 + · · ·+an : ai ∈ Ai for i = 1, 2, . . . , n}. If A and B arenonempty subsets of R letAB denote the set of all finite sums {a1b1+· · ·+anbn :n ∈ N∗; ai ∈ A; bi ∈ B}. More generally let A1A2 · · · , An denote the set ofall finite sums of elements of the form a1a2 . . . an (ai ∈ Ai for i = 1, 2, . . . , n).In the special case when all Ai(1 ≤ i ≤ n) are the same set A we denoteA1A2 · · ·An = AA · · ·A by An.

Theorem 3.2.6. Let A,A1, A2, . . . , An, B and C be (left) ideals in a ring R.(i) A1 + A2 + · · ·+ An and A1A2 · · ·An are (left) ideals;(ii) (A+B) + C = A+ (B + C);(iii) (AB)C = ABC = A(BC);(iv) B(A1 +A2 + · · ·+An) = BA1 +BA2 + · · ·BAn; and (A1 +A2 + · · ·+

An)C = A1C + A2C + · · ·+ AnC.

Ideals play approximately the same role in the theory of rings as normal sub-groups do in the theory of groups. For instance, let R be a ring and I an idealof R. Since the additive group of R is abelian, I is a normal subgroup. Conse-quently, there is a well-defined quotient group R/I in which addition is given by

(a+ I) + (b+ I) = (a+ b) + I

R/I can in fact be made into a ring.

Theorem 3.2.7. Let R be a ring and I an ideal of R. Then the additive quotientgroup R/I is a ring with multiplication given by

24

CHAPTER 3. RINGS

(a+ I)(b+ I) = ab+ I .

If R is commutative or has an identity, then the same is true of R/I .

Theorem 3.2.8. If f : R → S is a homomorphism of rings, then the kernel off is an ideal in R. Conversely, if I is an ideal in R, then the map π : R → R/Igiven by r 7→ r+I is an epimorphism of rings with kernel I. The map π is calledthe canonical epimorphism.

Theorem 3.2.9. If f : R → S is a homomorphism of rings and I is an ideal ofR which is contained in the kernel of f, then there is a unique homomorphism ofrings f : R/I → S such that f(a + I) = f(a) for all a ∈ R. Imf = Imf andker f = (ker f)/I . f is an isomorphism if and only if f is an epimorphism andI = ker f .

Corollary 3.2.10. (First Isomorphism Theorem) If f : R → S is a homomor-phism of rings, then f induces an isomorphism of rings R/ ker f ∼= Imf .

Corollary 3.2.11. If f : R → S is a homomorphism of rings, I is an ideal in Rand J is an ideal in S such that f(I) ⊂ J , then f induces a homomorphism ofrings f : R/I → S/J , given by a + I 7→ f(a) + J . f is an isomorphism if andonly if Imf + J = S and f−1(J) ⊂ I . In particular, if f is an epimorphism suchthat f(I) = J and ker f ⊂ I , then f is an isomorphism.

Theorem 3.2.12. Let I and J be ideals in a ring R.(i) (Second Isomorphism Theorem) There is an isomorphism of rings I/(I ∩

J) ∼= (I + J)/J;(ii) (Third Isomorphism Theorem) if I ⊂ J , then J/I is an ideal in R/I and

there is an isomorphism of rings (R/I)/(J/I) ∼= R/J .

Theorem 3.2.13. If I is an ideal in a ring R, then there is a one-to-one corre-spondence between the set of all ideals of R which contain I and the set of allideals of R/I , given by J 7→ J/I . Hence every ideal in R/I is of the form J/I ,where J is an ideal of R which contains I.

Definition 3.2.14. An ideal P in a ring R is said to be prime if P 6= R and forany ideals A,B in R

AB ⊂ P ⇒ A ⊂ P or B ⊂ P .

Theorem 3.2.15. If P is an ideal in a ring R such that P 6= R and for all a, b ∈ R

ab ∈ P ⇒ a ∈ P or b ∈ P ,

then P is prime. Conversely if P is prime and R is commutative, then P satisfiesprevious condition.

25

CHAPTER 3. RINGS

Theorem 3.2.16. In a commutative ring R with identity 1R 6= 0 an ideal P isprime if and only if the quotient ring R/P is an integral domain.

Definition 3.2.17. An ideal (resp. left ideal) M in a ring R is said to be maximalif M 6= R and for every ideal (resp. left ideal) N such that M ⊂ N ⊂ R, eitherN = M or N = R.

Theorem 3.2.18. In a nonzero ring R with identity, maximal (left) ideals alwaysexist. In fact, every (left) ideal in R (except R itself) is contained in a maximal(left) ideal.

Theorem 3.2.19. If R is a commutative ring with identity, then every maximalideal M in R is prime.

Theorem 3.2.20. Let M be an ideal in a ring R with identity 1R 6= 0.(i) If M is maximal and R is commutative, then the quotient ring R/M is a

field.(ii) If the quotient ring R/M is a division ring, then M is maximal.

Corollary 3.2.21. The following conditions on a commutative ring R with iden-tity 1R 6= 0 are equivalent.

(i) R is a field;(ii) R has no proper ideals;(iii) 0 is a maximal ideal in R;(iv) every nonzero homomorphism of rings R→ S is a monomorphism.

Theorem 3.2.22. Let {Ri : i ∈ I} be a nonempty family of rings and∏

i∈I Ri

the direct product of the additive abelian groups Ri;(i)∏

i∈I Ri is a ring with multiplication defined by {ai}i∈I{bi}i∈I = {aibi}i∈I;(ii) if Ri has an identity (resp. is comm) for every i ∈ I , then

∏i∈I Ri has an

identity (resp. is comm);(iii) for each k ∈ I the canonical projection πk :

∏i∈I Ri → Rk given by

{ai} 7→ ak, is an epimorphism of rings;(iv) for each k ∈ I the canonical injection ik : Rk →

∏i∈I Ri, given by

ak 7→ {ai} (where ai = 0 for i 6= k), is a monomorphism of rings.

Theorem 3.2.23. Let {Ri : i ∈ I} be a nonempty family of rings, S a ringand {φi : S → Ri : i ∈ I} a family of homomorphisms of rings. Then thereis a unique homomorphism of rings φ : S →

∏i∈I Ri such that πiφ = φi for

all i ∈ I . The ring∏

i∈I Ri is uniquely determined up to isomorphism by thisproperty.

Theorem 3.2.24. Let A1, A2, . . . , An be ideals in a ring R such that (i) A1 +A2 + · · · + An = R and (ii) for each k(1 ≤ k ≤ n), Ak ∩ (A1 + · · · + Ak−1 +Ak+1+· · ·+An) = 0. Then there is a ring isomorphismR ∼= A1×A2×· · ·×An.

If R is a ring andA1, . . . , An are ideals in R that satisfy previous theorem, thenR is said to be the internal direct product of the ideals Ai. Similar to groups,

26

CHAPTER 3. RINGS

if a ring R is the internal direct product of ideals, then each one is actually anideal contained in R, and R is isomorphic to the their direct product. However,the external direct product of A1 × · · · × An does not contain the Ai, but onlyisomorphic copies of them.

Definition 3.2.25. The ideals A and B of a ring R are said to be comaximal ifA+B = R.

Theorem 3.2.26. (Chinese Remainder Theorem) Let A1, . . . , Ak be ideals in aring R. The map R→ R/A1×R/A2× · · ·×R/Ak defined by r 7→ (r+A1, r+A2, . . . , r +Ak) is a ring homomorphism with kernel A1 ∩A2 ∩ · · · ∩Ak. If foreach i, j ∈ {1, 2, . . . , k} with i 6= j the ideals Ai and Aj are comaximal, thenthis map is surjective and A1 ∩ A2 ∩ · · · ∩ Ak = A1A2 · · ·Ak, so

R/(A1A2 · · ·Ak) = R/(A1 ∩ A2 ∩ · · · ∩ Ak) ∼= R/A1 ×R/A2 × · · · ×R/Ak.

Corollary 3.2.27. Letm1,m2, . . . ,mn be positive integers such that (mi,mj) =1 for i 6= j. If b1, b2, . . . , bn are any integers, then the system of congruencesx ≡ b1( mod m1);x ≡ b2( mod m2); · · · ;x ≡ bn( mod mn) has an integralsolution that is uniquely determined modulo m = m1m2 · · ·mn.

3.3 FACTORIZATION IN COMMUTATIVE RINGS

Definition 3.3.1. A nonzero element a of a commutative ring R is said to dividean element b ∈ R (notation:a|b) if there exists x ∈ R such that ax = b. Elementsa, b of R are said to be associates if a|b and b|a.

Theorem 3.3.2. Let a,b and u be elements of a commutative ring R with identity.(i) a|b if and only if (b) ⊂ (a).(ii) a and b are associates if and only if (a) = (b).(iii) u is a unit if and only if u|r for all r ∈ R.(iv) u is a unit if and only if (u) = R.(v) The relation “a is an associate of b” is an equivalence relation on R.(vi) If a = br with r ∈ R a unit, then a and b are associates. If R is an

integral domain, the converse is true.

Definition 3.3.3. Let R be a commutative ring with identity. An element c of Ris irreducible provided that:

(i) c is a nonzero nonunit;(ii) c = ab⇒ a or b is a unit.

An element p of R is prime provided that:(i) p is a nonzero nonunit;(ii) p|ab⇒ p|a or p|b.

27

CHAPTER 3. RINGS

Theorem 3.3.4. Let p and c be nonzero elements in an integral domain R.(i) p is prime if and only if (p) is nonzero prime ideal;(ii) c is irreducible if and only if (c) is maximal in the set S of all proper

principle ideals of R.(iii) Every prime element of R is irreducible.(iv) If R is a priciple ideal domain, then p is prime if and only if p is irre-

ducible.(v) Every associate of an irreducible (resp. prime) element of R is irreducible

(resp. prime).(vi) The only divisors of an irreducible element of R are its associates and the

units of R.

Definition 3.3.5. An integral domain R is a unique factorization domain pro-vided that:

(i) every nonzero nonunit element a of R can be written a = c1c2 · · · cn, withc1, . . . , cn irreducible.

(ii) If a = c1c2 · · · cn and a = d1d2 · · · dm (ci, di irreducible), then n = m andfor some permutation σ of {1, 2, . . . , n}, ci and dσ(i) are associates for every i.

Every irreducible element in a UFD is necessarily prime by (ii). Conse-quently, irreducible and prime elements coincide.

Lemma 3.3.6. If R is a principle ideal ring and (a1) ⊂ (a2) ⊂ · · · is a chain ofideals in R, then for some positive integer n, (aj) = (an) for all j ≥ n.

Theorem 3.3.7. Every principle ideal domain R is a unique factorization do-main.

Definition 3.3.8. Let N be the set of nonnegative integers and R a commutativering. R is a Euclidean ring if there is a function φ : R\{0} → N such that:

(i) if a, b ∈ R and ab 6= 0, then φ(a) ≤ φ(ab);(ii) if a, b ∈ R and b 6= 0, then there exists q, r ∈ R such that a = qb+ r with

r = 0, or r 6= 0 and φ(r) < φ(b).A Euclidean ring which is an integral domain is called a Euclidean domain.

Theorem 3.3.9. Every Euclidean ring R is a principle ideal ring with identity.Consequently, every Euclidean domain is a unique factorization domain.

Definition 3.3.10. Let X be a nonempty subset of a commutative ring R. Anelement d ∈ R is a greatest common divisor of X provided:

(i) d|a for all a ∈ X;(ii) c|a for all a ∈ X ⇒ c|d.

Greatest common divisors do not always exist. Also, if R has an identity anda1, a2, . . . , an have 1R as a greatest common divisor, then a1, a2, . . . , an are saidto be relatively prime.

28

CHAPTER 3. RINGS

Theorem 3.3.11. Let a1, . . . , an be elements of a commutative ring R with iden-tity.

(i) d ∈ R is a greatest common divisor of {a1, . . . , an} such that d = r1a1 +· · ·+ rnan for some ri ∈ R if and only if (d) = (a1) + (a2) + · · ·+ (an);

(ii) if R is a principle ideal ring, then a greatest common divisor of a1, . . . , anexists and every one is of the form r1a1 + · · ·+ rnan(ri ∈ R);

(iii) if R is a unique factorization domain, then there exists a greatest commondivisor of a1, . . . , an.

3.4 RINGS OF QUOTIENTS AND LOCALIZATION

Definition 3.4.1. A nonempty subset S of a ring R is multiplicative providedthat a, b ∈ S ⇒ ab ∈ S.

Theorem 3.4.2. Let S be a multiplicative subset of a commutative ring R. Therelation defined on the set R× S by

(r, s) ∼ (r′, s′)⇔ s1(rs′ − r′s) = 0 for some s1 ∈ S

is an equivalence relation. Furthermore if R has no zero divisors and 0 6∈ S,then

(r, s) ∼ (r′, s′)⇔ rs′ − r′s = 0.

Let S be a multiplicative subset of a commutative ring R and ∼ the equiv-alence relation of previous theorem. The equivalence class of (r, s) ∈ R × Swill be denoted r/s. The set of all equivalence classes of R × S under ∼ willbe denoted by S−1R. We have (i) r/s = r′/s′ ⇔ s1(rs

′ − r′s) = 0 for somes1 ∈ S. (ii) tr/ts = r/s for all r ∈ R and s, t ∈ S. (iii) If 0 ∈ S, then S−1Rconsists of a single equivalence class.

Theorem 3.4.3. Let S be a multiplicative subset of a commutative ring R and letS−1R be the set of equivalence classes of R× S under ∼.

(i) S−1R is a commutative ring with identity, where addition and multiplica-tion are defined by

r/s+ r′/s′ = (rs′ + r′s)/ss′ and (r/s)(r′/s′) = rr′/ss′.

(ii) If R is a nonzero ring with no zero divisors and 0 6∈ S, then S−1R is anintegral domain.

(iii) If R is a nonzero ring with no zero divisors and S is the set of all nonzeroelements of R, then S−1R is a field.

The ring S−1R is called the ring of quotients, ring of fractions, quotientring of R by S. An important case occurs when S is the set of all nonzero el-ements in an integral domain R. Then S−1R is called the quotient field of theintegral domain R. Thus if R = Z, the quotient field is precisely the field Q.

29

CHAPTER 3. RINGS

More generally suppose R is any nonzero commutative ring and S is the set ofall nonzero elements of R that are not zero divisors. If S is nonempty, then S−1Ris called the complete ring of quotients of the ring R.

Theorem 3.4.4. Let S be a multiplicative subset of a commutative ring R.(i) The map φS : R → S−1R given by r 7→ rs/s (for any s ∈ S) is a

well-defined homomorphism of rings such that φS(s) is a unit in S−1R for everys ∈ S.

(ii) If 0 6∈ S and S contains no zero divisors, then φS is a monomorphism. Inparticular, any integral domain may be embedded in its quotient field.

(iii) If R has an identity and S consists of units, then φS is an isomorphism.In particular, the complete ring of quotients (=quotient field) of a field F is iso-morphic to F.

Theorem 3.4.5. Let S be a multiplicative subset of a commutative ring R andlet T be any commutative ring with identity. If f : R → T is a homomorphismof rings such that f(s) is a unit in T for all s ∈ S, then there exists a uniquehomomorphism of rings f : S−1R → T such that fφS = f . The ring S−1R iscompletely determined (up to isomorphism) by this property.

Corollary 3.4.6. Let R be an integral domain considered as a subring of itsquotient field F. If E is a field and f : R → E a monomorphism of rings, thenthere is a unique monomorphism of fields f : F → E such that f |R = f . Inparticular any field E1 containing R contains an isomorphic copy F1 of F withR ⊂ F1 ⊂ E1.

Theorem 3.4.7. Let S be a multiplicative subset of a commutative ring R.(i) If I is an ideal in R, then S−1I = {a/s : a ∈ I; s ∈ S} is an ideal in

S−1R.(ii) If J is another ideal in R, then

S−1(I + J) = S−1I + S−1J;S−1(IJ) = (S−1I)(S−1J);S−1(I ∩ J) = S−1I ∩ S−1J .

S−1I is called the extension of I in S−1R. Note that r/s ∈ S−1I need notimply that r ∈ I since it is possible to have a/s = r/s with a ∈ I, r 6∈ I .

Theorem 3.4.8. Let S be a multiplicative subset of a commutative ring R withidentity and let I be an ideal of R. Then S−1I = S−1R if and only if S ∩ I 6= ∅.

Recall that if J is an ideal in a ring of quotients S−1R, then φ−1S (J) is an idealin R. It is sometimes called the contraction of J in R.

Lemma 3.4.9. Let S be a multiplicative subset of a commutative ring R withidentity and let I be an ideal in R.

(i) I ⊂ φ−1S (S−1I).

30

CHAPTER 3. RINGS

(ii) If I = φ−1S (J) for some ideal J in S−1R, then S−1I = J . In other wordsevery ideal in S−1R is of the form S−1I for some ideal I in R.

(iii) If P is a prime ideal in R and S ∩ P = ∅, then S−1P is a prime ideal inS−1R and φ−1S (S−1P ) = P .

Theorem 3.4.10. Let S be a multiplicative subset of a commutative ring R withidentity. Then there is a one-to-one correspondence between the set U of primeideals of R which are disjoint from S and the set V of prime ideals of S−1R,given by P 7→ S−1P .

Let R be a commutative ring with identity and P a prime ideal of R. ThenS = R\P is a multiplicative subset of R. The ring of quotients S−1R is calledthe localization of R at P and denoted RP . If I is an ideal in R, then the idealS−1I in RP is denoted IP .

Theorem 3.4.11. Let P be a prime ideal in a commutative ring R with identity.(i) There is a one-to-one correspondence between the set of prime ideals of R

which are contained in P and the set of prime ideals of RP , given by Q 7→ QP ;(ii) the ideal PP in RP is the unique maximal ideal of RP .

Definition 3.4.12. A local ring is a commutative ring with identity which has aunique maximal ideal.

Since every ideal in a ring with identity is contained in some maximal ideal,the unique maximal ideal of a local ring R must contain every ideal of R (exceptof course R itself). As an example, if p is prime and n ≥ 1, then Zpn is a localring with unique maximal ideal (p).

Theorem 3.4.13. If R is a commutative ring with identity then the followingconditions are equivalent.

(i) R is a local ring;(ii) all nonunits of R are contained in some ideal M 6= R;(iii) the nonunits of R form an ideal.

3.5 RINGS OF POLYNOMIALS AND FORMAL POWER SERIES

Theorem 3.5.1. Let R be a ring and let R[x] denote the set of all sequences ofelements of R (a0, a1, . . .) such that ai = 0 for all but a finite number of indicesi.

(i) R[x] is a ring with addition and multiplication defined by:

(a0, a1, . . .) + (b0, b1, . . .) = (a0 + b0, a1 + b1, . . .) and(a0, a1, . . .)(b0, b1, . . .) = (c0, c1, . . .), where

cn =∑n

i=0 an−ibi = anb0 + an−1b1 + · · ·+ a1bn−1 + a0bn =∑

k+j=n akbj .

31

CHAPTER 3. RINGS

(ii) If R is commutative (resp. a ring with identity or a ring with no zerodivisors or an integral domain), then so is R[x].

(iii) The map R → R[x] given by r 7→ (r, 0, 0, . . .) is a monomorphism ofrings.

If R has an identity 1R, then (1R, 0, 0, . . .) is an identity inR[x]. The ringR[x]is called the ring of polynomials over R. Its elements are called polynomials.We shall identify R with its isomorphic image in R[x] and write (r, 0, 0, . . .)simply as r. Note that r(a0, a1, . . .) = (ra0, ra1, . . .).

Theorem 3.5.2. Let R be a ring with identity and denote by x the element(0, 1R, 0, 0, . . .) of R[x].

(i) xn = (0, 0, . . . , 0, 1R, 0, . . .), where 1R is the (n+ 1)st coordinate.(ii) If r ∈ R, then for each n ≥ 0, rxn = xnr = (0, . . . , 0, r, 0, . . .), where r

is the (n+ 1)st coordinate.(iii) For every nonzero polynomial f inR[x] there exists an integer n ∈ N and

elements a0, . . . an ∈ R such that f = a0x0+a1x

1+· · ·+anxn. The integer n andelements ai are unique in the sense that f = b0x

0 + b1x1 + · · ·+ bmx

m(bi ∈ R)impies m ≥ n; ai = bi for i = 1, 2, . . . , n; and bi = 0 for n < i ≤ m.

Theorem 3.5.3. Let R be a ring and denote by R[x1, . . . , xn] the set of all func-tions f : Nn → R such that f(u) 6= 0 for at most a finite number of elements uof Nn.

(i) R[x1, . . . , xn] is a ring with addition and multiplication defined by

(f + g)(u) = f(u) + g(u) and (fg)(u) =∑

v+w=u f(v)g(w),

where f, g ∈ R[x1, . . . , xn] and u ∈ Nn.(ii) If R is commutative (resp. a ring with identity or a ring without zero

divisors or an integral domain), then so is R[x1, . . . , xn].(iii) The map R → R[x1, . . . , xn] given by r 7→ fr, where fr(0, . . . , 0) = r

and f(u) = 0 for all other u ∈ Nn, is a monomorphism of rings.

The ringR[x1, . . . , xn] is called the ring of polynomials in n indeterminatesover R.

Corollary 3.5.4. If φ : R → S is a homomorphism of commutative rings ands1, s2, . . . , sn ∈ S, then the mapR[x1, . . . , xn]→ S given by f 7→ φf(s1, . . . , sn)is a homomorphism of rings. This map is called the evaluation/substitution ho-momorphism.

Corollary 3.5.5. Let R be a commutative ring with identity and n a positve inte-ger. For each k (1 ≤ k < n) there are isomorphisms of ringsR[x1, . . . , xk][xk+1, . . . , xn]∼= R[x1, . . . , xn] ∼= R[xk+1, . . . , xn][x1, . . . , xk].

Proposition 3.5.6. Let R be a ring and denote by R [[x]] the set of all sequencesof elements of R (a0, a1, . . .).

(i) R [[x]] is a ring with addition and multiplication defined as before.

32

CHAPTER 3. RINGS

(ii) The polynomial ring R[x] is a subring of R [[x]].(iii) If R is commutative (resp. a ring with identity or a ring with no zero

divisors or an integral domain), then so is R [[x]].

The ring R [[x]] is called the ring of formal power series over the ring R. Itselements are called power series.

Proposition 3.5.7. Let R be a ring with identity and f =∑∞

i=0 aixi ∈ R [[x]].

(i) f is a unit in R [[x]] if and only if its constant term a0 is a unit in R.(ii) If a0 is irreducible in R, then f is irreducible in R [[x]].

Corollary 3.5.8. If R is a division ring, then the units in R [[x]] are preciselythose power series with nonzero constant term. The principle ideal (x) consistsprecisely of the nonunits in R [[x]] and is the unique maximal ideal of R [[x]].Thus if R is a field, R [[x]] is a local ring.

3.6 FACTORIZATION IN POLYNOMIAL RINGS

Theorem 3.6.1. Let R be a ring and f, g ∈ R[x1, . . . , xn].(i) deg(f + g) ≤ max(deg f, deg g).(ii) deg(fg) ≤ deg f + deg g.(iii) If R has no zero divisors, deg(fg) = deg f + deg g.(iv) If n = 1 and the leading coefficient of f or g is not a zero divisor in R (in

particular, if it is not a unit), then deg(fg) = deg f + deg g.

Theorem 3.6.2. (The Division Algorithm) Let R be a ring with identity andf, g ∈ R[x] nonzero polynomials such that the leading coefficient of g is a unitin R. Then there exist unique polynomials q, r ∈ R[x] such that

f = qg + r and deg r < deg g.

Corollary 3.6.3. (Remainder Theorem) Let R be a ring with identity and

f(x) =∑n

i=0 aixi ∈ R[x].

For any c ∈ R there exists a unique q(x) ∈ R[x] such that f(x) = q(x)(x− c) +f(c).

Corollary 3.6.4. If F is a field, then the polynomial ring F [x] is a Euclidean do-main, hence F [x] is a principle ideal domain and a unique factorization domain.The units in F [x] are precisely the nonzero constant polynomials.

Definition 3.6.5. Let R be a subring of a commutative ring S, c1, c2, . . . , cn ∈ Sand f =

∑mi=0 aix

ki11 · · ·xkinn ∈ R[x1, . . . , xn] a polynomial such that f(c1, c2, . . . cn) =

0. Then (c1, c2, . . . , cn) is said to be a root/zero of f (or a solution of the poly-nomial equation f(x1, . . . , xn) = 0).

33

CHAPTER 3. RINGS

Theorem 3.6.6. Let R be a commutative ring with identity and f ∈ R[x]. Thenc ∈ R is a root of f if and only if x− c divides f.

Theorem 3.6.7. If D is an integral domain contained in an integral domain Eand f ∈ D[x] has degree n, then f has at most n distinct roots in E.

Proposition 3.6.8. Let D be a unique factorization domain with quotient field Fand let f =

∑ni=0 aix

i ∈ D[x]. If u = c/d ∈ F with c and d relatively prime,and u is a root of f, then c|a0 and d|an.

Let D be an integral domain and f ∈ D[x]. If c ∈ D and c is a root of f, thenthere is a greatest integer m (0 ≤ m ≤ deg f) such that f(x) = (x − c)mg(x),where g(x) ∈ R[x] and x − c - g(x) (that is, g(c) 6= 0). The integer m is calledthe multiplicity of the root c of f. If c has multiplicity 1, c is said to be a simpleroot. If c has multiplicity m > 1, c is called a multiple root.

Lemma 3.6.9. Let D be an integral domain and f =∑n

i=0 aixi ∈ D[x]. Let

f ′ ∈ D[x] be the polynomial f ′ =∑n

k=1 kakxk−1 = a1 + 2a2x+ 3a3x

2 + · · ·+nanx

n−1. Then for all f, g ∈ D[x] and c ∈ D:(i) (cf)′ = cf ′;(ii) (f + g)′ = f ′ + g′;(iii) (fg)′ = f ′g + fg′;(iv) (gn)′ = ngn−1g′.

The polynomial f’ is called the formal derivative of f. A nonzero polynomialf ∈ R[x] is irreducible provided f is not a unit and in every factorization f =gh, either g or h is a unit in R[x].

Theorem 3.6.10. Let D be an integral domain which is a subring of an integraldomain E. Let f ∈ D[x] and c ∈ E.

(i) c is a multiple root of f if and only if f(c) = 0 and f ′(c) = 0.(ii) If D is a field and f is relatively prime to f’, then f has no multiple roots in

E.(iii) If D is a field, f is irreducible in D[x] and E contains a root of f then f has

no multiple roots in E if and only if f ′ 6= 0.

We now consider the more general question of determining the units andirreducible elements in the polynomial ringD[x], where D is an integral domain.

(i) The units in D[x] are precisely the constant polynomials that are units inD.

(ii) If c ∈ D and c is irreducible in D, then the constant polynomial c isirreducible in D[x].

(iii) Every first degree polynomial whose leading coefficient is a unit in Dis irreducible in D[x]. In particular, every first degree polynomial over a field isirreducible.

(iv) Suppose D is a subring of an integral domain E and f ∈ D[x] ⊂ E[x].Then f may be irreducible in E[x] but not in D[x] and vice versa.

34

CHAPTER 3. RINGS

Let D be a UFD and f =∑n

i=0 aixi a nonzero polynomial in D[x]. A gcd

of the coefficients a0, a1, . . . , an is called a content of f and is denoted C(f). IfC(f) is a unit in D, then f is said to be primitive.

Lemma 3.6.11. (Gauss) If D is a unique factorization domain and f, g ∈ D[x],then C(fg) = C(f)C(g). In particular, the product of primitive polynomials isprimitive.

Lemma 3.6.12. Let D be a unique factorization domain with quotient field Fand let f and g be primitive polynomials in D[x]. Then f and g are associates inD[x] if and only if they are associates in F [x].

Lemma 3.6.13. Let D be a unique factorization domain with quotient field Fand f a primitive polynomial of positive degree in D[x]. Then f is irreducible inD[x] if and only if f is irreducible in F [x].

Theorem 3.6.14. If D is a unique factorization domain, then so is the polynomialring D[x1, . . . , xn].

Theorem 3.6.15. (Eisenstein’s Criterion) Let D be a unique factorization do-main with quotient field F. If f =

∑ni=0 aix

i ∈ D[x], deg f ≥ 1 and p is anirreducible element of D such that

p - an; p | ai for i = 0, 1, . . . , n− 1; p2 - a0,

then f is irreducible in F [x]. If f is primitive, then f is irreducible in D[x].

35

CHAPTER 3. RINGS

36

Chapter 4

Modules

4.1 MODULES, HOMOMORPHISMS AND EXACT SEQUENCES

Definition 4.1.1. Let R be a ring. A (left) R-module is an additive abelian groupA together with a function R×A→ A (the image of (r, a) being denoted by ra)such that for all r, s ∈ R and a, b ∈ A:

(i) r(a+ b) = ra+ rb.(ii) (r + s)a = ra+ sa.(iii) r(sa) = (rs)a.

If R has an identity element 1R and(iv) 1Ra = a for all a ∈ A,

then A is said to be a unitary R-module. If R is a division ring, then a unitaryR-module is called a (left) vector space.

A (unitary) right R-module is defined similarly via a function A × R → Adenoted (a, r) 7→ ar and satisfying the obvious analogues of (i)-(iv). Fromnow on, “R-module” will mean “left R-module” and it is understood that alltheorems about the left R-modules also hold for the right. Every module A of acommutative ring R is assumed to be both a left and right module with ar = rafor all r ∈ R, a ∈ A. For some examples, every additive abelian group G is aunitary Z-module. If S is a ring and R a subring, then S is an R-module. If I is aleft ideal of a ring R, then I is a left R-module.

Definition 4.1.2. Let A and B be modules over a ring R. A function f : A → Bis an R-module homomorphism provided that for all a, c ∈ A and r ∈ R:

f(a+ c) = f(a) + f(c) and f(ra) = rf(a).

If R is a division ring, then an R-module homomorphism is called a linear trans-formation.

Definition 4.1.3. Let R be a ring, A an R-module and B a nonempty subset of A.B is a submodule of A provided that B is an additive subgroup of A and rb ∈ Bfor all r ∈ R, b ∈ B. A submodule of a vector space over a division ring iscalled a subspace.

CHAPTER 4. MODULES

For example, if R is a ring and f : A → B is an R-module homomorphism,then ker f is a submodule of A and Im f is a submodule of B. If C is any sub-module of B, then f−1(C) = {a ∈ A : f(a) ∈ C} is a submodule of A.

Definition 4.1.4. If X is a subset of a module A over a ring R, then the intersec-tion of all submodules of A containing X is called the submodule generated byX (or spanned by X).

Theorem 4.1.5. Let R be a ring, A an R-module, X a subset of A, {Bi : i ∈ I} afamily of submodules of A and a ∈ A. Let Ra = {ra : r ∈ R}.

(i) Ra is a submodule of A and the map R → Ra given by r 7→ ra is anR-module epimorphism.

(ii) The cyclic submodule C generated by a is {ra + na : r ∈ R;n ∈ Z}. IfR has an identity and C is unitary, then C = Ra.

(iii) The submodule D generated by X is{s∑i=1

riai +t∑

j=1

njbj : s, t ∈ N∗; ai, bj ∈ X; ri ∈ R;nj ∈ Z

}

If R has an identity and A is unitary, then

D = RX =

{s∑i=1

riai : s ∈ N∗; ai ∈ X; ri ∈ R

}.

(iv) The sum of the family {Bi : i ∈ I} consists of all finite sums bi1 + · · ·+binwith bik ∈ Bik .

Theorem 4.1.6. Let B be a submodule of a module A over a ring R. Then thequotient group A/B is an R-module with the actoin of R on A/B given by:

r(a+B) = ra+B for all r ∈ R, a ∈ A.

The map π : A → A/B given by a 7→ a + B is an R-module epimorphism withkernel B. The map π is called the canonical epimorphism (or projection).

Theorem 4.1.7. If R is a ring and f : A → B is an R-module homomorphismand C is a submodule of ker f , then there is a unique R-module homomorphismf : A/C → B such that f(a + C) = f(a) for all a ∈ A; Im f =Im f andker f = ker f/C. f is an R-module isomorphism if and only if f is an R-moduleepimorphism and C = ker f . In particular, A/ ker f ∼= Im f.

Corollary 4.1.8. If R is a ring and A’ is a submodule of the R-module A and B’a submodule of the R-module B and f : A→ B is an R-module homomorphismsuch that f(A′) ⊂ B′, then f induces an R-module homomorphism f : A/A′ →B/B′ given by a+ A′ 7→ f(a) +B′. f is an R-module isomorphism if and onlyif Im f + B′ = B and f−1(B′) ⊂ A′. In particular if f is an epimorphism suchthat f(A′) = B′ and ker f ⊂ A′, then f is an R-module isomorphism.

38

CHAPTER 4. MODULES

Theorem 4.1.9. Let B and C be submodules of a module A over a ring R.(i) There is an R-module isomorphism B/(B ∩ C) ∼= (B + C)/C;(ii) if C ⊂ B, then B/C is a submodule of A/C, and there is an R-module

isomorphism (A/C)/(B/C) ∼= A/B.

Theorem 4.1.10. If R is a ring and B is a submodule of an R-module A, thenthere is a one-to-one correspondence between the set of all submodules of Acontaining B and the set of all submodules of A/B, given by C 7→ C/B. Henceevery submodule ofA/B is of the form C/B, where C is a submodule of A whichcontains B.

Theorem 4.1.11. Let R be a ring and {Ai : i ∈ I} a nonempty family of R-modules,

∏i∈I Ai the direct product of the abelian groups Ai, and

∑i∈I Ai the

direct sum of the abelian groups Ai.(i)∏

i∈I Ai is an R-module with the action of R given by r{ai} = {rai}.(ii)∑

i∈I Ai is a submodule of∏

i∈I Ai.(iii) For each k ∈ I , the canonical projection πk :

∏Ai → Ak is an R-module

epimorphism.(iv) For each k ∈ I , the canonical injection ik : Ak →

∑Ai is an R-module

monomorphism.

Theorem 4.1.12. Let R be a ring and A,A1, A2, . . . , An R-modules. Then A ∼=A1 ⊕ A2 ⊕ · · · ⊕ An if and only if for each i = 1, 2, . . . , n there are R-modulehomomorphisms πi : A→ Ai and ιi : Ai → A such that

(i) πiιi = 1Aifor i = 1, 2, . . . , n;

(ii) πjιi = 0 for i 6= j;(iii) ι1π1 + ι2π2 + · · ·+ ιnπn = 1A.

Theorem 4.1.13. Let R be a ring and {Ai : i ∈ I} a family of submodules of anR-module A such that

(i) A is the sum of the family {Ai : i ∈ I};(ii) for each k ∈ I, Ak ∩A∗k = 0, where A∗k is the sum of the family {Ai : i 6=

k}. Then there is an isomorphism A ∼=∑

i∈I Ai.

A module A is said to be the internal direct sum of a family of submod-ules {Ai : i ∈ I} provided that A and {Ai} satisfy the hypotheses of previoustheorem. We write A =

∑i∈I Ai to indicate internal direct sum.

Definition 4.1.14. A pair of module homomorphisms,Af−→ B

g−→ C, is said to beexact at B provided Im f = ker g. A finite sequence of module homomorphisms,

A0f1−→ A1

f2−→ A2f3−→ · · · fn−1−−→ An−1

fn−→ An, is exact provided Im fi = ker fi+1

for i = 1, 2, . . . , n−1. An infinite sequence of module homomorphisms, · · · fi−1−−→Ai−1

fi−→ Aifi+1−−→ Ai+1

fi+2−−→ · · · is exact provided Im fi = ker fi+1 for all i ∈ Z.

Note that for any module A, there are unique module homomorphisms 0→ A

and A→ 0. If A and B are any modules then the sequences 0→ Aι−→ A⊕B π−→

39

CHAPTER 4. MODULES

B → 0 and 0→ Bι−→ A⊕B π−→ A→ 0 are exact, where the ι’s and π’s are the

canonical injections and projections respectively. Similarly, if C is a submoduleof D, then the sequence 0 → C

i−→ Dp−→ D/C → 0 is exact where i is the

inclusion map and p the canonical epi. If f : A→ B is a module homomorphismthen A/ ker f (resp. B/Imf ) is called the coimage of f (resp. cokernel of f)and denoted Coim f (resp. Coker f). Each of the following sequences is exact:0 → ker f → A → Coimf → 0, 0 → Imf → B → Cokerf → 0 and0 → ker f → A

f−→ B → Cokerf → 0, where the unlabled maps are theobvious inclusions and projections.

0 → Af−→ B is an exact sequence of module homomorphisms if and only if

f is a module monomorphism. Similarly, Bg−→ C → 0 is exact if and only if g

is a module epimorphism. If Af−→ B

g−→ C is exact, then gf = 0. Finally, ifA

f−→ Bg−→ C → 0 is exact, then Coker f = B/Imf = B/ ker g = Coim g∼= C.

An exact sequence of the form 0 → Af−→ B

g−→ C → 0 is called a short exactsequence; note that f is a monomorphism and g an epimorphism. The precedingremarks show that a short exact sequence is just another way of presenting asubmodule (A ∼= Imf ) and its quotient module (B/Imf = B/ ker g ∼= C)

Lemma 4.1.15. (The Short Five Lemma) Let R be a ring and

0 A0 B C 0

0 A′ B′ C ′ 0

α β γ

f g

f ′ g′

a commutative diagram of R-modules and R-module homomorphisms such thateach row is a short exact sequence. Then

(i) α, γ monomorphisms⇒ β is a monomorphism;(ii) α, γ epimorphisms⇒ β is an epimorphism;(iii) α, γ isomorphisms⇒ β is an isomorphism.

Two short exact sequences are said to be isomorphic if there is a commutativediagram of module homomorphisms

0 A B C 0

0 A′ B′ C ′ 0

f g h

Also, isomorphism of short exact sequences is an equivalence relation.

Theorem 4.1.16. Let R be a ring and 0 → A1f−→ B

g−→ A2 → 0 a shortexact sequence of R-module homomorphisms. Then the following conditions areequivalent.

40

CHAPTER 4. MODULES

(i) There is an R-module homomorphism h : A2 → B with gh = 1A2;(ii) There is an R-module homomorphism k : B → A1 with kf = 1A1;(iii) the given sequence is isomorphic (with identity maps on A1 and A2) to

the direct sum short exact sequence 0 → A1ι1−→ A1 ⊕ A2

π2−→ A2 → 0; inparticular B ∼= A1 ⊕ A2.

A short exact sequence that satisfies these conditions is said to be split or asplit exact sequence.

4.2 FREE MODULES AND VECTOR SPACES

A subset X of an R-module A is said to be linearly independent provided thatfor distinct x1, . . . , xn ∈ X and ri ∈ R:

r1x1 + r2x2 + · · ·+ rnxn = 0⇒ ri = 0 for every i.

A set that is not linearly independent is said to be linearly dependent. If A isgenerated as an R-module by a set Y, then we say that Y spans A. If R has anidentity and A is unitary, Y spans A if and only if every element of A may bewritten as a linear combination: r1y1 + r2y2 + · · · + rnyn(ri ∈ R, yi ∈ Y ). Alinearly independent subset of A that spans A is called a basis of A.

Theorem 4.2.1. Let R be a ring with identity. The following conditions on aunitary R-module F are equivalent:

(i) F has a nonempty basis;(ii) F is the internal direct sum of a family of cyclic R-modules, each of which

is isomorphic as a left R-module to R;(iii) F is R-module isomorphic to a direct sum of copies of the left R-module

R;(iv) there exists a nonempty set X and a function ι : X → F with the following

property: given any unitary R-module A and function f : X → A, there exists aunique R-module homomorphism f : F → A such that fι = f .

A unitary module F over a ring R with identity, which satisfies these condi-tions is called a free R-module on the set X.

Corollary 4.2.2. Every (unitary) module A over a ring R (with identity) is thehomomorphic image of a free R-module F. If A is finitely generated, then F maybe chosen to be finitely generated.

Lemma 4.2.3. A maximal linearly independent subset X of a vector space V overa division ring D is a basis of V

Theorem 4.2.4. Every vector space V over a division ring D has a basis and istherefore a free D-module. More generally, every linearly independent subset ofV is contained in a basis of V

41

CHAPTER 4. MODULES

Theorem 4.2.5. If V is a vector space over a division ring D and X is a subsetthat spans V, then X contains a basis of V.

Theorem 4.2.6. Let R be a ring with identity and F a free R-module with aninfinite basis X. Then every basis of F has the same cardinality as X.

Theorem 4.2.7. If V is a vector space over a division ring D, then any two basesof V have the same cardinality.

Definition 4.2.8. Let R be a ring with identity such that for every free R-moduleF, any two bases of F have the same cardinality. Then R is said to have theinvariant dimension property and the cardinal number of any basis of F iscalled the dimension/rank of F over R.

The dimension of a vector space V over a division ring D will be denoted bydimDV .

Proposition 4.2.9. Let E and F be free modules over a ring R that has the in-variant dimension property. Then E ∼= F if and only if E and F have the samerank.

Proposition 4.2.10. Let f : R → S be a nonzero epimorphism of rings withidentity. If S has the invariant dimension property, then so does R.

Corollary 4.2.11. If R is a ring with identity that has a homomorphic imagewhich is a division ring, then R has the invariant dimension property. In partic-ular, every commutative ring with identity has the invariant dimension property.

A vector space V over a division ring D is said to be finite dimensional ifdimDV is finite.

Theorem 4.2.12. Let W be a subspace of a vector space V over a division ringD.

(i) dimDW ≤ dimDV ;(ii) if dimDW = dimDV and dimDV is finite, then W = V ;(iii) dimDV = dimDW+ dimD(V/W ).

Corollary 4.2.13. If f : V → V ′ is a linear transformation of vector spacesover a division ring D, then there exists a basis X of V such that X ∩ ker f is abasis of ker f and {f(x) : f(x) 6= 0, x ∈ X} is a basis of Im f. In particular,

dimDV = dimD(ker f)+ dimD(Im f).

Corollary 4.2.14. If V and W are finite dimensional subspaces of a vector spaceover a division ring D, then

dimDV+dimDW =dimD(V ∩W )+dimD(V +W ).

Theorem 4.2.15. Let R,S,T be division rings such that R ⊂ S ⊂ T . Then

dimRT = (dimST )(dimRS).

Furthermore, dimRT is finite if and only if dimST and dimRS are finite.

42

CHAPTER 4. MODULES

4.3 PROJECTIVE AND INJECTIVE MODULES

Definition 4.3.1. A module P over a ring R is said to be projective if given anydiagram of R-module homomorphisms

P

A B 0

f

g

with bottom row exact (that is, g an epimorphism), there exists an R-modulehomomorphism h : P → A such that the diagram

P

A B 0

f

g

h

is commutative (that is, gh = f ).

Theorem 4.3.2. Every free module F over a ring R with identity is projective.

Corollary 4.3.3. Every module A over a ring R is the homomorphic image of aprojective R-module.

Theorem 4.3.4. Let R be a ring. The following conditions on an R-module P areequivalent.

(i) P is projective;(ii) every short exact sequence 0 → A

f−→ Bg−→ P → 0 is split exact (hence

B ∼= A⊕ P );(iii) there is a free module F and an R-module K such that F ∼= K ⊕ P .

For an example, if R = Z6, then Z3 and Z2 are Z6-modules and there is a Z6-isomorphism Z6

∼= Z2 ⊕ Z3. Hence both Z2 and Z3 are projective Z6-modulesthat are not free Z6-modules.

Proposition 4.3.5. Let R be a ring. A direct sum of R-modules∑

i∈I Pi is pro-jective if and only if each Pi is projective.

Definition 4.3.6. A module J over a ring R is said to be injective if given anydiagram of R-module homomorphisms

0 A B

J

g

f

43

CHAPTER 4. MODULES

with top row exact (that is, g a monomorphism), there exists an R-module homo-morphism h : B → J such that the diagram

0 A B

J

g

f h

is commutative (that is, hg = f ).

Proposition 4.3.7. A direct product of R-modules∏

i∈I Ji is injective if and onlyif Ji is injective for every i ∈ I .

Lemma 4.3.8. Let R be a ring with identity. A unitary R-module J is injective ifand only if for every left ideal L of R, any R-module homomorphism L→ J maybe extended to an R-module homomorphism R→ J .

An abelian group D is said to be divisible if given any y ∈ D and 0 6= n ∈ Z,there exists x ∈ D such that nx = y. For example, the additive group Q isdivisible, but Z is not.

Lemma 4.3.9. An abelian group D is divisible if and only if D is an injective(unitary) Z-module.

Lemma 4.3.10. Every abelian group A may be embedded in a divisible abeliangroup.

Lemma 4.3.11. If J is a divisible abelian group and R is a ring with identity,then HomZ(R, J) is an injective left R-module.

Proposition 4.3.12. Every unitary module A over a ring R with identity may beembedded in an injective R-module.

Proposition 4.3.13. Let R be a ring with identity. The following conditions on aunitary R-module J are equivalent.

(i) J is injective;(ii) every short exact sequence 0 → J

f−→ Bg−→ C → 0 is split exact (hence

B ∼= J ⊕ C);(iii) J is a direct summand of any module B of which it is a submodule.

4.4 HOM AND DUALITY

Recall that if A and B are modules over a ring R, then HomR(A,B) is the setof all R-module homomorphisms f : A → B. If R = Z we shall usuallywrite Hom(A,B) in place of HomZ(A,B). HomR(A,B) is an abelian group

44

CHAPTER 4. MODULES

under addition and this addition is distributive with respect to composition offunctions.

Theorem 4.4.1. Let A,B,C,D be modules over a ring R and φ : C → A and ψ :B → D R-module homomorphisms. Then the map θ :HomR(A,B)→HomR(C,D)given by f 7→ ψfφ is a homomorphism of abelian groups.

The map θ is usually denoted Hom(φ, ψ) and called the homomorphism in-duced by φ and ψ. Observe that for homomorphisms φ1 : E → C, φ2 : C →A,ψ1 : B → D,ψ2 : D → F ,

Hom(φ1, ψ2)Hom(φ2, ψ1) =Hom(φ2φ1, ψ2ψ1) :HomR(A,B)→HomR(E,F ).

Theorem 4.4.2. Let R be a ring. 0 → Aφ−→ B

ψ−→ C is an exact sequence ofR-modules if and only if for every R-module D

0→ HomR(D,A)φ−→ HomR(D,B)

ψ−→ HomR(D,C)

is an exact sequence of abelian groups.

Proposition 4.4.3. Let R be a ring. A θ−→ Bζ−→ C → 0 is an exact sequence of

R-modules if and only if for every R-module D

0→ HomR(C,D)ζ−→ HomR(B,D)

θ−→ HomR(A,D)

is an exact sequence of abelian groups.

One sometimes summarizes the two preceding results by saying thatHomR(A,B)is left exact. It is not true in general that a short exact sequence 0→ A→ B →C → 0 induces a short exact sequence 0 → HomR(D,A) → HomR(D,B) →HomR(D,C)→ 0. However the next three theorems show that this result doeshold in several cases.

Proposition 4.4.4. The following conditions on modules over a ring R are equiv-alent.

(i) 0→ Aφ−→ B

ψ−→ C → 0 is a split exact sequence of R-modules;

(ii) 0 → HomR(D,A)φ−→ HomR(D,B)

ψ−→ HomR(D,C) → 0 is a splitexact sequence of abelian groups for every R-module D;

(iii) 0 → HomR(C,D)ψ−→ HomR(B,D)

φ−→ HomR(A,D) → 0 is a splitexact sequence of abelian groups for every R-module D;

Theorem 4.4.5. The following conditions on a module P over a ring R are equiv-alent

(i) P is projective;(ii) if ψ : B → C is any R-module epimorphism then ψ : HomR(P,B) →

HomR(P,C) is an epimorphism of abelian groups;

(iii) if 0→ Aφ−→ B

ψ−→ C → 0 is any short exact sequence of R-modules, then

0→ HomR(P,A)φ−→ HomR(P,B)

ψ−→ HomR(P,C)→ 0 is an exact sequenceof abelian groups.

45

CHAPTER 4. MODULES

Proposition 4.4.6. The following conditions on a module J over a ring R areequivalent.

(i) J is injective;(ii) if θ : A → B is any R-module monomorphism then θ : HomR(B, J) →

HomR(A, J) is an epimorphism of abelian groups;

(iii) if 0→ Aθ−→ B

ζ−→ C → 0 is any short exact sequence of R-modules, then

0→ HomR(C, J)ζ−→ HomR(B, J)

θ−→ HomR(A, J)→ 0 is an exact sequenceof abelian groups.

Theorem 4.4.7. Let A,B,{Ai : i ∈ I} and {Bj : j ∈ J} be modules over a ringR. Then there are isomorphisms of abelian groups:

(i) HomR(∑

i∈I Ai, B) ∼=∏

i∈I HomR(Ai, B);(ii) HomR(A,

∏j∈J Bj) ∼=

∏j∈J HomR(A,Bj).

Let R and S be rings. An abelian group A is an R-S bimodule provided thatA is both a left R-module and a right S-module and

r(as) = (ra)s for all a ∈ A, r ∈ R, s ∈ S.

We sometimes write RAS to indicate the fact that A is an R-S bimodule. Simi-larly RB indicates a left R-module B and CS a right S-module C.

Theorem 4.4.8. Let R and S be rings and let RA,RBS,R CS,RD be (bi)modulesas indicated.

(i) HomR(A,B) is a right S-module, with the action of S given by (fs)(a) =(f(a))s(s ∈ S; a ∈ A; f ∈ HomR(A,B)).

(ii) If φ : A → A′ is a homomorphism of left R-modules, then the inducedmap φ : HomR(A′, B)→ HomR(A,B) is a homomorphism of right S-modules.

(iii) HomR(C,D) is a left S-module, with the action of S given by (sg)(c) =g(cs)(s ∈ S; c ∈ C; g ∈ HomR(C,D)).

(iv) Ifψ : D → D′ is a homomorphism of left R-modules, thenψ : HomR(C,D)→HomR(C,D′) is a homomorphism of left S-modules.

Theorem 4.4.9. If A is a unitary left module over a ring R with identity thenthere is an isomorphism of left R-modules A ∼= HomR(R,A).

Let A be a left module over a ring R. Since R is an R-R bimodule,HomR(A,R)is a right R-module. HomR(A,R) is called the dual module of A and is denotedA∗. The elements of A∗ are sometimes called linear functionals.

Theorem 4.4.10. Let A,B and C be left modules over a ring R.(i) If φ : A→ C is a homomorphism of left R-modules, then the induced map

φ : C∗ = HomR(C,R) → HomR(A,R) = A∗ is a homomorphism of rightR-modules.

(ii) There is an R-module isomorphism (A⊕ C)∗ ∼= A∗ ⊕ C∗.(iii) If R is a division ring and 0 → A

θ−→ Bζ−→ C → 0 is a short exact

sequence of left vector spaces, then 0 → C∗ζ−→ B∗

θ−→ A∗ → 0 is a short exactsequence of right vector spaces.

46

CHAPTER 4. MODULES

The map φ of (i) is called the dual map of φ. Kronecker delta notation: forany set I and ring R with identity the symbol δij(i, j ∈ I) denotes 0 ∈ R if i 6= jand 1R ∈ R if i = j

Theorem 4.4.11. Let F be a free left module over a ring R with identity. Let X bea basis of F and for each x ∈ X let fx : F → R be given by fx(y) = δxy(y ∈ X).Then

(i) {fx : x ∈ X} is a linearly independent subset of F ∗ of cardinality |X|;(ii) if X is finite, then F ∗ is a free right R-module with basis {fx : x ∈ X}.

In part (ii) {fx : x ∈ X} is called the dual basis to X.

Theorem 4.4.12. Let A be a left module over a ring R.(i) There is an R-module homomorphism θ : A→ A∗∗.(ii) If R has an identity and A is free, then θ is a monomorphism.(iii) If R has an identity and A is free with a finite basis, then θ is an isomor-

phism.

A module A such that θ : A→ A∗∗ is an isomorphism is said to be reflexive.

4.5 TENSOR PRODUCTS

The tensor product A ⊗R B of modules AR and RB over a ring R is a certainabelian group. If C is an (additive) abelian group, then a middle linear mapfromA×B to C is a function f : A×B → C such that for all a, ai ∈ A, b, bi ∈ Band r ∈ R:

f(a1 + a2, b) = f(a1, b) + f(a2, b);f(a, b1 + b2) = f(a, b1) + f(a, b2);

f(ar, b) = f(a, rb).

Definition 4.5.1. Let A be a right module and B a left module over a ring R. Let Fbe the free abelian group on the setA×B. Let K be the subgroup of F generatedby all elements of the following forms (for all a, a′ ∈ A; b, b′ ∈ B; r ∈ R):

(i) (a+ a′, b)− (a, b)− (a′, b);(ii) (a, b+ b′)− (a, b)− (a, b′);(iii) (ar, b)− (a, rb).

The quotient group F/K is called the tensor product of A and B; it is denotedA⊗R B (or simply A⊗B if R = Z). The coset (a, b) +K of the element (a, b)in F is denoted a⊗ b; the coset (0, 0) is denoted 0.

Given modules AR and RB over a ring R, it is easy to verify that the mapι : A×B → A⊗R B given by (a, b) 7→ a⊗ b is a middle linear map. This mapis called the canonical middle linear map.

47

CHAPTER 4. MODULES

Theorem 4.5.2. Let AR and RB be modules over a ring R, and C be an abeliangroup. If g : A × B → C is a middle linear map, then there exists a uniquegroup homomorphism g : A ⊗R B → C such that gi = g, where i : A × B →A⊗RB is the canonical middle linear map. A⊗RB is uniquiely determined upto isomorphism by this property.

Corollary 4.5.3. If AR, A′R,RB and RB′ are modules over a ring R and f :

A → A′, g : B → B′ are R-module homomorphisms, then there is a uniquegroup homomorphism A⊗R B → A′ ⊗R B′ such that a⊗ b 7→ f(a)⊗ g(b) forall a ∈ A, b ∈ B.

Proposition 4.5.4. If Af−→ B

g−→ C → 0 is an exact sequence of left modulesover a ring R and D is a right R-module, then

D ⊗R A1D⊗f−−−→ D ⊗R B

1D⊗g−−−→ D ⊗R C → 0

is an exact sequence of abelian groups. An analogous statement holds for anexact sequence in the first variable.

Theorem 4.5.5. Let R and S be rings and SAR,RB,CR,RDS (bi)modules asindicated.

(i) A⊗R B is a left S-module such that s(a⊗ b) = sa⊗ b for all s ∈ S, a ∈A, b ∈ B.

(ii) If f : A → A′ is a homomorphism of S-R bimodules and g : B → B′ isan R-module homomorphism, then the induced map f⊗g : A⊗RB → A′⊗RB′is a homomorphism of left S-modules.

(iii) C⊗RD is a right S-module such that (c⊗d)s = c⊗ds for all c ∈ C, d ∈D, s ∈ S.

(iv) If h : C → C ′ is an R-module homomorphism and k : D → D′ ahomomorphism of R-S bimodules, then the induced map h ⊗ k : C ⊗R D →C ′ ⊗R D′ is a homomorphism of right S-modules.

Let A,B,C be modules over a commutative ring R. A bilinear map fromA × B to C is a function f : A × B → C such that for all a, ai ∈ A, b, bi ∈ B,and r ∈ R:

f(a1 + a2, b) = f(a1, b) + f(a2, b);f(a, b1 + b2) = f(a, b1) + f(a, b2);f(ra, b) = rf(a, b) = f(a, rb).

Theorem 4.5.6. If A,B,C are modules over a commutative ring R and g : A ×B → C is a bilinear map, then there is a unique R-module homomorphismg : A⊗RB → C such that gi = g, where i : A×B → A⊗RB is the canonicalbilinear map. The module A⊗RB is uniquely determined up to isomorphism bythis property.

Theorem 4.5.7. If R is a ring with identity and AR,RB are unitary R-modules,then there are R-module isomorphisms

48

CHAPTER 4. MODULES

A⊗R R ∼= A and R⊗R B ∼= B.

Theorem 4.5.8. If R and S are rings and AR,RBS,S C are (bi)modules, thenthere is an isomorphism

(A⊗R B)⊗S C ∼= A⊗R (B ⊗S C).

Theorem 4.5.9. Let R be a ring, A and {Ai : i ∈ I} right R-modules, B and{Bj : j ∈ J} left R-modules. Then there are group isomorphisms:

(∑

i∈I Ai)⊗R B ∼=∑

i∈I(Ai ⊗R B);A⊗R (

∑j∈J Bj) ∼=

∑j∈J(A⊗R Bj).

Theorem 4.5.10. (Adjoint Associativity) Let R and S be rings and AR,RBS, CS(bi)modules. Then there is an isomorphism of abelian groups

α : HomS(A⊗R B,C) ∼= HomR(A,HomS(B,C)),

defined for each f : A⊗R B → C by

[(αf)(a)](b) = f(a⊗ b).

Theorem 4.5.11. Let R be a ring with identity. If A is a unitary right R-moduleand F is a free left R-module with basis Y, then every element u of A⊗R F maybe written uniquely in the form u =

∑ni=1 ai ⊗ yi, where ai ∈ A and the yi are

distinct elements of Y.

Theorem 4.5.12. If R is a ring with identity and AR and RB are free R-moduleswith bases X and Y respectively, then A ⊗R B is a free (right) R-module withbasis W = {x⊗ y : x ∈ X, y ∈ Y } of cardinality |X||Y |.

Corollary 4.5.13. Let S be a ring with identity and R a subring of S that contains1S . If F is a free left R-module with basis X, then S ⊗R F is a free left S-modulewith basis {1S ⊗ x : x ∈ X} of cardinality |X|.

4.6 MODULES OVER A PRINCIPLE IDEAL DOMAIN

Throughout this section, “module” will mean “unitary module”

Theorem 4.6.1. Let F be a free module over a principle ideal domain R and Ga submodule of F. Then G is a free R-module and rank G ≤ rank F.

Corollary 4.6.2. Let R be a principle ideal domain. If A is a finitely generatedR-module generated by n elements, then every submodule of A may be generatedby m elements with m ≤ n.

Corollary 4.6.3. A unitary module A over a principle ideal domain is free if andonly if A is projective.

49

CHAPTER 4. MODULES

Theorem 4.6.4. Let A be a left module over an integral domain R and for eacha ∈ A let Oa = {r ∈ R : ra = 0}.

(i) Oa is an ideal of R for each a ∈ A.(ii) At = {a ∈ A : Oa 6= 0} is a submodule of A.(iii) For each a ∈ A there is an isomorphism of left modules

R/Oa ∼= Ra = {ra : r ∈ R}.Let R be a principle ideal domain and p ∈ R a prime.

(iv) If pia = 0 (equivalently (pi) ⊂ Oa), then Oa = (pj) with 0 ≤ j ≤ i.(v) If Oa = (pi), then pja 6= 0 for all j such that 0 ≤ j < i.

Let A be a module over an integral domain. The ideal Oa is called the orderideal of a ∈ A. The submodule At is called the torsion submodule of A. A issaid to be a torsion module if A = At and to be torsion free if At = 0. Everyfree module is torsion-free. For A a module over a PID R, the order ideal ofa ∈ A is a principle ideal of R, say Oa = (r), and a is said to have order r. Theelement r is unique only up to multiplication by a unit. The cyclic submoduleRa generated by a is said to be cyclic of order r.a ∈ A has order 0 if andonly if Ra ∼= R. Also, a has order r, with r a unit, if and only if a = 0; (fora = 1Ra = r−1(ra) = r−10 = 0).

Theorem 4.6.5. A finitely generated torsion-free module A over a principle idealdomain R is free.

Theorem 4.6.6. If A is a finitely generated module over a principle ideal domainR, then A = At ⊕ F , where F is a free R-module of finite rank and F ∼= A/At.

Theorem 4.6.7. Let A be a torsion module over a principle ideal domain R andfor each prime p ∈ R let A(p) = {a ∈ A : a has order a power of p}.

(i) A(p) is a submodule of A for each prime p ∈ R;(ii) A =

∑A(p), where the sum is over all primes p ∈ R. If A if finitely

generated, only finitely many of the A(p) are nonzero.

Lemma 4.6.8. Let A be a module over a principle ideal domain R such thatpnA = 0 and pn−1A 6= 0 for some prime p ∈ R and positive integer n. Let a bean element of A of order pn.

(i) If A 6= Ra, then there exists a nonzero b ∈ A such that Ra ∩Rb = 0.(ii) There is a submodule C of A such that A = Ra⊕ C.

Theorem 4.6.9. Let A be a finitely generated module over a principle ideal do-main R such that every element of A has order a power of some prime p ∈ R.Then A is a direct sum of cycic R-modules of order pn1 , . . . , pnk respectively,where n1 ≥ n2 ≥ · · · ≥ nk ≥ 1.

Lemma 4.6.10. Let A,B andAi(i ∈ I) be modules over a principle ideal domainR. Let r ∈ R and let p ∈ R be prime.

(i) rA = {ra : a ∈ A} and A[r] = {a ∈ A : ra = 0} are submodules of A.(ii) R/(p) is a field and A[p] is a vector space over R/(p).(iii) For each positive integer n there are R-module isomorphisms

50

CHAPTER 4. MODULES

(R/(pn))[p] ∼= R/(p) and pm(R/(pn)) ∼= R/(pn−m)(0 ≤ m < n).

(iv) If A ∼=∑

i∈I Ai, then rA ∼=∑

i∈I rAi and A[r] ∼=∑

i∈I Ai[r].(v) If f : A → B is an R-module isomorphism, then f : At ∼= Bt and

A(p) ∼= B(p).

Lemma 4.6.11. Let R be a principle ideal domain. If r ∈ R factors as r =pn11 · · · p

nkk with p1, . . . , pk ∈ R distinct primes and each ni > 0, then there is an

R-module isomorphism

R/(r) ∼= R/(pn11 )⊕ · · · ⊕R/(pnk

k ).

Consequently every cyclic R-module of order r is a direct sum of k cyclic R-modules of orders pn1

1 , . . . , pnkk respectively.

Theorem 4.6.12. Let A be a finitely generated module over a principle idealdomain R.

(i) A is the direct sum of a free submodule F of finite rank and a finite numberof cyclic torsion modules. The cyclic torsion summands (if any) are of ordersr1, . . . , rt, where r1, . . . , rt are (not necessarily distinct) nonzero nonunit ele-ments of R such that r1|r2| · · · |rt. The rank of F and the list of ideals (r1), . . . , (rt)are uniquely determined by A.

(ii) A is the direct sum of a free submodule E of finite rank and a finite num-ber of cycic torsion modules. The cyclic torsion summands (if any) are of or-ders ps11 , . . . , p

skk , where p1, . . . , pk are (not necessarily distinct) primes in R and

s1, . . . , sk are (not necessarily distinct) positive integers. The rank of E and thelist of ideals (ps11 ), . . . , (pskk ) are uniquely determined by A (except for the orderof the pi).

The elements r1, . . . , rt are called the invariant factors of the module A.Similarly ps11 , . . . , p

skk are called the elementary divisors of A.

Corollary 4.6.13. Two finitely generated modules over a principle ideal domain,A and B, are isomorphic if and only if A/At and B/Bt have the same rank andA and B have the same invariant factors (resp. elementary divisors).

4.7 ALGEBRAS

Definition 4.7.1. Let K be a commutative ring with identity. A K-algebra/algebraover K A is a ring A such that:

(i) (A,+) is a unitary (left) K-module;(ii) k(ab) = (ka)b = a(kb) for all k ∈ K and a, b ∈ A.A K-algebra A which, as a ring, is a division ring, is called a division alge-

bra.

51

CHAPTER 4. MODULES

An algebra over a field K that is finite dimensional as a vector space over K iscalled a finite dimensional algebra over K. Every ring R is an additive abeliangroup and hence a Z-module. It is easy to see that R is actually a Z-algebra. LetG be a multiplicative group and K a commutative ring with identity. Then thegroup ring K(G) is actually a K-algebra with K-module structure given by

k(∑rigi) =

∑(kri)gi (k, ri ∈ K; gi ∈ G).

K(G) is called the group algebra of G over K.

Theorem 4.7.2. Let K be a commutative ring with identity and A a unitary leftK-module. Then A is a K-algebra if and only if there exists a K-module homo-morphism π : A⊗K A→ A such that the diagram

A⊗K A⊗K A A⊗K A

A⊗K A A

π⊗1A

1A⊗π π

π

is commutative. In this case the K-algebra A has an identity if and only if thereis a K-module homomorphism I : K → A such that the diagram

K ⊗K A A A⊗K K

A⊗K A A A⊗K A

ζ

I⊗1A 1A 1A⊗I

θ

ππ

is commutative.

The homomorphism π is called the product map of the K-algebra A. Thehomomorphism I is called the unit map.

Definition 4.7.3. Let K be a commutative ring with identity and A, B K-algebras.(i) A subalgebra of A is a subring of A that is also a K-submodule of A.(ii) A (left, right, two-sided) algebra ideal of A is a (left, right, two-sided)

ideal of the ring A that is also a K-submodule of A.(iii) A homomorphism (resp. isomorphism) of K-algebras f : A → B is

a ring homomorphism (isomorphism) that is also a K-module homomorphism(isomorphism).

Theorem 4.7.4. Let A and B be algebras (with identity) over a commutative ringK with identity. Let π be the composition

(A⊗K B)⊗K (A⊗K B)1A⊗α⊗1B−−−−−−→ (A⊗K A)⊗K (B ⊗K B)

πA⊗πB−−−−→ A⊗K B,

where πA, πB are the product maps of A and B respectively. Then A ⊗K B is aK-algebra (with identity) with product map π.

The K-algebra A ⊗K B of previous theorem is called the tensor product ofthe K-algebras A and B.

52

Chapter 5

Fields and Galois Theory

5.1 FIELD EXTENSIONS

Definition 5.1.1. A field F is said to be an extension field of K (or simply anextension of K) provided that K is a subfield of F.

F is a vector space over K. The dimension of the K-vector space F will bedenoted by [F : K] rather than dimKF . F is said to be a finite dimensionalextension or infinite dimensional extension of K according as [F : K] is finiteor infinite.

Theorem 5.1.2. Let F be an extension field of E and E and extension field of K.Then [F : K] = [F : E][E : K]. Furthermore [F : K] is finite if and only if[F : E] and [E : K] are finite.

In the situation K ⊂ E ⊂ F , E is said to be an intermediate field of Kand F. If F is a field and X ⊂ F , then the subfield (resp. subring) generatedby X is the intersection of all subfields (resp. subrings) of F that contain X.If F is an extension field of K and X ⊂ F , then the subfield (resp. subring)generated by K ∪X is called the subfield (resp. subring) generated by X overK and is denoted K(X) (resp. K[X]). Note that K[X] is necessarily an integraldomain. The field K(u1, . . . , un) is said to be a finitely generated extensionof K (but it need not be finite dimensional over K). If X = {u} then K(u) issaid to be a simple extension of K. We also have that K(u1, . . . , un−1)(un) =K(u1, . . . , un) and K[u1, . . . , un−1][un] = K[u1, . . . , un].

Theorem 5.1.3. If F is an extension field of a field K, u, ui ∈ F , and X ⊂ F ,then

(i) the subring K[u] consists of all elements of the form f(u), where f is apolynomial with coefficients in K (that is, f ∈ K[x]);

(ii) the subringK[u1, . . . , um] consists of all elements of the form g(u1, u2, . . . , um),where g is a polynomial in m indeterminates with coefficients in K (that is,g ∈ K[x1, . . . , xm]);

CHAPTER 5. FIELDS AND GALOIS THEORY

(iii) the subring K[X] consists of all elements of the form h(u1, . . . , un)where each ui ∈ X , n is a positive integer, and h is a polynomial in n inde-terminates with coefficients in K (that is, n ∈ N∗, h ∈ K[x1, . . . , xn]);

(iv) the subfield K(u) consists of all elements of the form f(u)/g(u) =f(u)g(u)−1, where f, g ∈ K[x] and g(u) 6= 0;

(v) the subfield K(u1, . . . , um) consists of all elements of the form

h(u1, . . . , um)/k(u1, . . . , um) = h(u1, . . . , um)k(u1, . . . , um)−1,

where h, k ∈ K[x1, . . . , xm] and k(u1, . . . , um) 6= 0;(vi) the subfield K(X) consists of all elements of the form

f(u1, . . . , un)/g(u1, . . . , un) = f(u1, . . . , un)g(u1, . . . , un)−1

where n ∈ N∗, f, g ∈ K[x1, . . . , xn], u1, . . . , un ∈ X and g(u1, . . . , un) 6= 0.(vii) For each v ∈ K(X) (resp. K[X]) there is a finite subset of X ′ of X such

that v ∈ K(X ′) (resp. K[X ′]).

If L and M are subfields of a field F, the composite of L and M in F, denotedLM is the subfield generated by the set L ∪M . Also, LM = L(M) = M(L).

Definition 5.1.4. Let F be an extension field of K. An element u of F is said to bealgebraic over K provided that u is a root of some nonzero polynomial f ∈ K[x].If u is not a root of any nonzero f ∈ K[x], u is said to be transcendental overK. F is called an algebraic extension of K if every element of F is algebraicover K. F is called a transcendental extension if at least one element of F istranscendental over K.

As an example, i ∈ C is algebraic over Q but π, e ∈ R are transcendentalover Q.

The quotient field of K[x1, . . . , xn] is denoted K(x1, . . . , xn). It consists ofall fractions f/g, with f, g ∈ K[x1, . . . , xn] and g 6= 0 and is called the field ofrational functions in x1, . . . , xn over K.

Theorem 5.1.5. If F is an extension field of K and u ∈ F is transcendental overK, then there is an isomorphism of fields K(u) ∼= K(x) which is the identity onK.

Theorem 5.1.6. If F is an extension field of K and u ∈ F is algebraic over K,then

(i) K(u) = K[u];(ii) K(u) ∼= K[x]/(f), where f ∈ K[x] is an irreducible monic polynomial

of degree n ≥ 1 uniquely determined by the condition that f(u) = 0 and g(u) =0(g ∈ K[x]) if and only if f divides g;

(iii) [K(u) : K] = n;(iv) {1K , u, u2, . . . , un−1} is a basis of the vector space K(u) over K;(v) every element of K(u) can be written uniquely in the form a0 + a1u +

· · ·+ an−1un−1(ai ∈ K).

54


Definition 5.1.7. Let F be an extension field of K and u ∈ F algebraic over K.The monic irreducible polynomial f from previous theorem, is called the irre-ducible/minimal polynomial of u. The degree of u over K is deg f = [K(u) :K].

Theorem 5.1.8. Let σ : K → L be an isomorphism of field, u an element ofsome extension field of K and v an element of some extension field of L. Assumeeither

(i) u is transcendental over K and v is transcendental over L; or(ii) u is a root of an irreducible polynomial f ∈ K[x] and v is a root of

σf ∈ L[x].Then σ extends to an isomorphism of fields K(u) ∼= L(v) which maps u onto v.

Corollary 5.1.9. Let E and F each be extension fields of K and let u ∈ E and v ∈F be algebraic over K. then u and v are roots of the same irreducible polynomialf ∈ K[x] if and only if there is an isomorphism of fields K(u) ∼= K(v) whichsends u onto v and is the identity on K.

Theorem 5.1.10. If K is a field and f ∈ K[x] a polynomial of degree n, thenthere exists a simple extension field F = K(u) of K such that:

(i) u ∈ F is a root of f;(ii) [K(u) : K] ≤ n, with equality holding if and only if f is irreducible in

K[x];(iii) if f is irreducible in K[x], then K(u) is unique up to an isomorphism

which is the identity on K.

Theorem 5.1.11. If F is a finite dimensional extension field of K, then F is finitelygenerated and algebraic over K.

Theorem 5.1.12. If F is an extension field of K and X is a subset of F such thatF = K(X) and every element of X is algebraic over K, then F is an algebraicextension of K. If X is a finite set, then F is finite dimensional over K.

Theorem 5.1.13. If F is an algebraic extension field of E and E is an algebraicextension field of K, then F is an algebraic extension of K.

Theorem 5.1.14. Let F be an extension field of K and E the set of all elementsof F which are algebraic over K. Then E is a subfield of F (which is, of course,algebraic over K).

5.2 THE FUNDEMENTAL THEOREM

The main idea of Galois Theory is to consider the relation of the group of per-mutations of the roots of f(x) to the algebraic structure of its splitting field.

55


Definition 5.2.1. Let E and F be extension fields of a field K. A nonzero mapσ : E → F which is both a field and a K-module homomorphism is calleda K-homomorphism. Similarly if a field automorphism σ ∈ AutF is a K-homomorphism, then σ is called a K-automorphism of F. The group of allK-automorphisms of F is called the Galois group of F over K and is denotedAutKF or Gal(F/K).

Note that any field has at least one automorphism, the identity map, denotedby 1 and sometimes called the trivial automorphism. The prime field of K isgenerated by 1 ∈ K and since any automorphism σ takes 1 to 1 (and 0 to 0),i.e., σ(1) = 1, it follows that σa = a for all a in the prime field. Hence anyautomorphism of a field K fixes its prime subfield.

Theorem 5.2.2. Let F be an extension field of K and f ∈ K[x]. If u ∈ F is aroot of f and σ ∈ AutKF , then σ(u) ∈ F is also a root of f.

When u is algebraic over K with irreducible polynomial f ∈ K[x] of degreen, then any σ ∈ AutKK(u) is completely determined by its action on u (since{1K , u, u2, . . . , un−1} is a basis of K(u) over K). Since σ(u) is a root of f, then|AutKK(u)| ≤ m, where m is the number of distinct roots of f in K(u); (m ≤n).

Theorem 5.2.3. Let F be an extension field of K, E an intermediate field and Ha subgroup of AutKF . Then

(i) H ′ = {v ∈ F : σ(v) = v for all σ ∈ H} is an intermediate field of theextension;

(ii) E ′ = {σ ∈ AutKF : σ(u) = u for all u ∈ E} = AutEF is a subgroupof AutKF .

The field H ′ is called the fixed field of H in F. If we denote AutKF by G, itis not necessarily true that G′ = K.

Proposition 5.2.4. The association of groups to fields and fields to groups de-fined above is inclusion reversing, namely

(i) if F1 ⊂ F2 ⊂ K are two subfield of K then Aut(K/F2) ≤ Aut(K/F1),and

(ii) if H1 ≤ H2 ≤ Aut(K) are two subgoups of automorphisms with associ-ated fixed fields F1 and F2, respectively, then F2 ⊂ F1.

Definition 5.2.5. Let F be an extension field of K such that the fixed field of theGalois group AutKF is K itself. Then F is said to be a Galois extension (field)of K or to be Galois over K.

F is Galois over K if and only if for any u ∈ F − K, there exists a K-automorphism σ ∈ AutKF such that σ(u) 6= u. If F is an arbitrary extensionfield of K and K0 is the fixed field of AutKF (possibly K0 6= K), then it is easyto see that F is Galois over K0, that K ⊂ K0, and that AutKF = AutK0F . IfL,M are intermediate fields of an extension with L ⊂M , the dimension [M : L]

56


is called the relative dimension of L and M. Similarly, if H,J are subgroups ofthe Galois group with H < J , the index [J : H] is called the relative index ofH and J.

Theorem 5.2.6. (Fundemental Theorem of Galois Thoery) If F is a finite dimen-sional Galois extension of K, then there is a one-to-one correspondence betweenthe set of all intermediate fields of the extension and the set of all subgroups ofthe Galois group AutKF (given by E 7→ E ′ = AutEF ) such that:

(i) the relative dimension of two intermediate fields is equal to the relativeindex of the corresponding subgroups; in particular, AutKF has order [F : K];

(ii) F is Galois over every intermediate field E, but E is Galois over K if andonly if the corresponding subgroup E ′ = AutEF is normal in G = AutKF ; inthis case G/E ′ is (isomorphic to) the Galois group AutKE of E over K.

Theorem 5.2.7. (Artin) Let F be a field, G a group of automorphisms of F andK the fixed field of G in F. Then F is Galois over K. If G is finite, then F is a finitedimensional Galois extension of K with Galois group G.

5.3 SPLITTING FIELDS, ALGEBRAIC CLOSURE AND

NORMALITY

Let F be a field and f ∈ F [x] a polynomial of positive degree. f is said to splitover F (or to split in F [x]) if f can be written as a product of linear factors inF [x]; that is, f = u0(x− u1)(x− u2) · · · (x− un) with ui ∈ F .

Definition 5.3.1. Let K be a field and f ∈ K[x] a polynomial of positive degree.An extension field F of K is said to be a splitting field over K of the polynomialf if f splits in F [x] and F = K(u1, . . . , un) where u1, . . . , un are the roots of f inF.

Let S be a set of polynomials of positive degree in K[x]. An extension field Fof K is said to be a splitting field over K of the set S of polynomials if everypolynomial in S splits in F [x] and F is generated over K by the roots of all thepolynomials in S

Note that if u is a root of an irreducible f ∈ K[x], K(u) need not be a splittingfield of f. If F is a splitting field of S over K, then F = K(X), where X is theset of all roots of polynomials in the subset S of K[x].

Theorem 5.3.2. If K is a field and f ∈ K[x] has degree n ≥ 1, then there existsa splitting field F of with [F : K] ≤ n!.

Theorem 5.3.3. The following conditions on a field F are equivalent.(i) Every nonconstant polynomial f ∈ F [x] has a root in F;(ii) every nonconstant polynomial f ∈ F [x] splits over F;

57


(iii) every irreducible polynomial in F [x] has degree one;(iv) there is no algebraic extension field of F (except F itself);(v) there exists a subfield K of F such that F is algebraic over K and every

polynomial in K[x] splits in F [x].

A field that satisfies these conditions is said to be algebraically closed.

Theorem 5.3.4. If F is an extension field of K, then the following conditions areequivalent.

(i) F is algebraic over K and F is algebraically closed;(ii) F is a splitting field over K of the set of all (irreducible) polynomials in

K[x].

An extension field F of a field K that satisfies these conditions is called analgebraic closure of K. For example, C = R(i) is an algebraic closure of R.

Theorem 5.3.5. Every field K has an algebraic closure. Any two algebraic clo-sures of K are K-isomorphic.

Corollary 5.3.6. If K is a field and S a set of polynomials (of positive degree) inK[x], then there exists a splitting field of S over K.

Theorem 5.3.7. Let σ : K → L be an isomorphism of fields, S = {fi} a set ofpolynomials (of positive degree) in K[x], and S ′ = {σfi} the corresponding setof polynomials in L[x]. If F is a splitting field of S over K and M is a splittingfield of S ′ over L, then σ is extendible to an isomorphism F ∼= M .

Corollary 5.3.8. Let K be a field and S a set of polynomials (of positive de-gree) in K[x]. Then any two splitting fields of S over K are K-isomorphic. Inparticular, any two algebraic closures of K are K-isomorphic.

Definition 5.3.9. Let K be a field and f ∈ K[x] an irreducible polynomial. Thepolynomial f is said to be separable if in some splitting field of f over K everyroot of f is a simple root.

If F is an extension field of K and u ∈ F is algebraic over K, then u is said tobe separable over K provided its irreducible polynomial is separable. If everyelement of F is sperable over K, then F is said to be a separable extension of K.

It is clear that a seperable polynomial f ∈ K[x] has no multiple roots in anysplitting field of f over K. Also an irreducible polynomial in K[x] is seperable ifchar K = 0. Hence every algebraic extension field of a field of characteristic 0is separable.

Theorem 5.3.10. If F is an extension field of K, then the following statementsare equivalent.

(i) F is algebraic and Galois over K;(ii) F is separable over K and F is a splitting field over K of a set S of poly-

nomials in K[x];(iii) F is a splitting field over K of a set T of separable polynomials in K[x].

58


Definition 5.3.11. An algebraic extension field F of K is normal over K or anormal extension if every irreducible polynomial in K[x] that has a root in Factually splits in F [x].

Theorem 5.3.12. If F is an algebraic extension field of K, then the followingstatements are equivalent.

(i) F is normal over K;(ii) F is a splitting field over K of some set of polynomials in K[x];(iii) ifK is any algebraic closure of K containing F, then for any K-monomorphism

of fields σ : F → K, Im σ = F so that σ is actually a K-automorphism of F.

Corollary 5.3.13. Let F be an algebraic extension field of K. Then F is Galoisover K if and only if F is normal and separable over K. If char K = 0, then F isGalois over K if and only if F is normal over K.

Theorem 5.3.14. If E is an algebraic extension field of K, then there exists anextension field F of E such that

(i) F is normal over K;(ii) no proper subfield of F containing E is normal over K;(iii) if E is separable over K, then F is Galois over K;(iv) [F : K] is finite if and only if [E : K] is finite.

The field F is uniquely determined up to an E-isomorphism.

The field F is sometimes called the normal closure of E over K.

5.4 THE GALOIS GROUP OF A POLYNOMIAL

Definition 5.4.1. Let K be a field. The Galois group of a polynomial f ∈ K[x]is the group AutKF , where F is a splitting field of f over K.

Recall that a subgroup G of Sn is said to be transitive if given any i 6= j(1 ≤i, j ≤ n), there exists σ ∈ G such that σ(i) = j.

Theorem 5.4.2. Let K be a field and f ∈ K[x] a polynomial with Galois groupG.

(i) G is isomorphic to a subgroup of some symmetric group Sn.(ii) If f is (irreducible) seperable of degree n, then n divides |G| and G is

isomorphic to a transitive subgoup of Sn.

Corollary 5.4.3. Let K be a field and f ∈ K[x] an irreducible polynomial ofdegree 2 with Galois group G. If f is separable (as is always the case when charK 6= 2), then G ∼= Z2; otherwise G = 1.

Note that the Galois group of a separable polynomial of degree 3 is either S3

or A3 (the only transitive subgroup of S3).

59


Definition 5.4.4. Let K be a field with char K 6= 2 and f ∈ K[x] a polynomialof degree n with n distinct roots u1, . . . , un in some splitting field F of f overK. Let ∆ =

∏i<j(ui − uj) = (u1 − u2)(u1 − u3) · · · (un−1 − un) ∈ F ; the

discriminant of f is the element D = ∆2.

Proposition 5.4.5. Let K, f, F and ∆ be as before, then(i) The discriminant ∆2 of f actually lies in K.(ii) For each σ ∈ AutKF < Sn, σ is an even (resp. odd) permutation if and

only if σ(∆) = ∆ (resp. σ(∆) = −∆).

Corollary 5.4.6. Let K, f, F, ∆ be as before, (so that F is Galois over K) andconsider G = AutKF as a subgroup of Sn. In the Galois correspondence, thesubfield K(∆) corresponds to the subgroup G∩An. In particular, G consists ofeven permutations if and only if ∆ ∈ K.

Corollary 5.4.7. Let K be a field and f ∈ K[x] an (irreducible) separable poly-nomial of degree 3. The Galois group of f is either S3 or A3. If char K 6= 2, it isA3 if and only if the discriminant of f is the square of an element of K.

Proposition 5.4.8. Let K be a field with char K 6= 2, 3. If f(x) = x3 + bx2 +cx+d ∈ K[x] has three distinct roots in some splitting field, then the polynomialg(x) = f(x− b/3) ∈ K[x] has the form x3 + px+ q and the discriminant of f is−4p3 − 27q2.

Theorem 5.4.9. If p is prime and f is an irreducible polynomial of degree p overthe field of rational numbers which has precisely two nonreal roots in the field ofcomplex numbers, then the Galois group of f is (isomorphic to) Sp.

5.5 FINITE FIELDS

Theorem 5.5.1. Let F be a field and let P be the intersection of all subfields of F.Then P is a field with no proper subfields. If char F = p (prime), then P ∼= Zp.If char F = 0, then P ∼= Q, the field of rational numbers.

The field P is called the prime subfield of F.

Corollary 5.5.2. If F is a finite field, then char F = p 6= 0 for some prime p and|F | = pn for some integer n ≥ 1.

Theorem 5.5.3. If F is a field and G is a finite subgroup of the multiplicativegroup of nonzero elements of F, then G is a cyclic group. In particular, themultiplicative group of all nonzero elements of a finite field is cyclic.

Corollary 5.5.4. If F is a finite field, then F is a simple extension of its primesubfield Zp; that is, F = Zp(u) for some u ∈ F .

60


Lemma 5.5.5. If F is a field of characteristic p and r ≥ 1 is an integer, then themap φ : F → F given by u 7→ up

ris a Zp-monomorphism of fields. If F is finite,

then φ is a Zp-automorphism of F.

Proposition 5.5.6. Let p be a prime and n ≥ 1 an integer. Then F is a finite fieldwith pn elements if and only if F is a splitting field of xp

n − x over Zp.

Corollary 5.5.7. If p is a prime and n ≥ 1 an integer, then there exists a fieldwith pn elements. Any two finite fields with the same number of elements areisomorphic.

Corollary 5.5.8. If K is a finite field and n ≥ 1 is an integer, then there exists asimple extension field F = K(u) of K such that F is finite and [F : K] = n. Anytwo n-dimensional extension fields of K are K-isomorphic.

Corollary 5.5.9. If K is a finite field and n ≥ 1 an integer, then there exists anirreducible polynomial of degree n in K[x].

Proposition 5.5.10. If F is a finite dimensional extension field of a finite field K,then F is finite and is Galois over K. The Galois group AutKF is cyclic.

5.6 SEPARABILITY

Definition 5.6.1. Let F be an extension field of K. An algebraic element u ∈ F ispurely inseparable over K if its irreducible polynomial f in K[x] factors in F [x]as f = (x− u)m. F is a purely inseparable extension of K if every element ofF is purely inseparable over K.

Thus u is separable over K if its irreducible polynomial f of degree n hasn distinct roots (in some splitting field) and purely inseparable over K if f hasprecisely one root.

Theorem 5.6.2. Let F be an extension field of K. Then u ∈ F is both separbleand purely inseparable over K if and only u ∈ K.

We have the following fact: if charK = p 6= 0 and u, v ∈ K, then (u±v)pn

=up

n ± vpn for all n ≥ 0.

Lemma 5.6.3. Let F be an extension field of K with char K = p 6= 0. If u ∈ Fis algebraic over K, then up

nis separable over K for some n ≥ 0.

Theorem 5.6.4. If F is an algebraic extension field of a field K of characteristicp 6= 0, then the following statements are equivalent:

(i) F is purely inseparable over K;(ii) the irreducible polynomial of any u ∈ F is of the form xp

n − a ∈ K[x];(iii) if u ∈ F , then up

n ∈ K for some n ≥ 0;

61


(iv) the only elements of F which are separable over K are the elements of Kitself;

(v) F is generated over K by a set of purely inseparable elements.

Corollary 5.6.5. If F is a finite dimensional purely inseparable extension fieldof K and char K = p 6= 0, then [F : K] = pn for some n ≥ 0.

Lemma 5.6.6. If F is an extension field of K, X is a subset of F such thatF = K(X), and every element of X is separable over K, then F is a separa-ble extension of K.

Theorem 5.6.7. Let F be an algebraic extension field of K, S the set of all ele-ments of F which are separable over K, and P the set of all elements of F whichare purely inseparable over K.

(i) S is a separable extension field of K.(ii) F is purely inseparable over S.(iii) P is a purely inseparable extension field of K.(iv) P ∩ S = K.(v) F is separable over P if and only if F = SP .(vi) If F is normal over K, then S is Galois over K, F is Galois over P and

AutKS ∼= AutPF = AutKF .

Corollary 5.6.8. If F is a separable extension field of E and E is a separableextension field of K, then F is separable over K.

Corollary 5.6.9. Let F be an algebraic extension field of K, with char K = p 6=0. If F is separable over K, then F = KF pn for each n ≥ 1. If [F : K] is finiteand F = KF p, then F is separable over K. In particular, u ∈ F is separableover K if and only if K(up) = K(u).

Definition 5.6.10. Let F be an algebraic extension field of K and S the largestsubfield of F separable over K. The dimension [S : K] is called the separabledegree of F over K and is denoted [F : K]s. The dimension [F : S] is calledthe inseparable degree/degree of inseparability of F over K and is denoted[F : K]i.

Lemma 5.6.11. Let F be an extension field of E, E an extension field of Kand N a normal extension field of K containing F. If r is the cardinal numberof distinct E-monomorphisms F → N and t is the cardinal number of dis-tinct K-monomorphisms E → N , then rt is the cardinal number of distinctK-monomorphisms F → N .

Proposition 5.6.12. Let F be a finite dimensional extension field of K and N anormal extension field of K containing F. The number of distinct K-monomorphismsF → N is precisely [F : K]s, the separable degree of F over K.

Corollary 5.6.13. If F is an extension field of E and E is an extension field of K,then

62


[F : E]s[E : K]s = [F : K]s and [F : E]i[E : K]i = [F : K]i

Corollary 5.6.14. Let f ∈ K[x] be an irreducible monic polynomial over a fieldK, F a splitting field of f over K and u1 a root of f in F. Then

(i) every root of f has multiplicity [K(u1) : K]i so that in F [x],

f = [(x− u1) · · · (x− un)][K(u1):K]i ,

where u1, . . . , un are all distinct roots of f and n = [K(u1) : K]s;(ii) u[K(u1):K]i

1 is separable over K.

Proposition 5.6.15. (Primitive Element Theorem) Let F be a finite dimensionalextension field of K.

(i) If F is separable over K, then F is a simple extension of K.(ii) (Artin) More generally, F is a simple extension of K if and only if there

are only finitely may intermediate fields.

An element u such that F = K(u) is said to be primitive.

5.7 CYCLIC EXTENSIONS

Definition 5.7.1. Let F be a finite dimensional extension field of K and K an al-gebraic closure of K containing F. Let σ1, . . . , σr be all the distinct K-monomorphismsF → K. If u ∈ F , the norm of u, denoted, NF

K(u) is the element

NFK(u) = (σ1(u)σ2(u) · · · σr(u))[F :K]i .

The trace of u, denoted T FK (u), is the element

T FK (u) = [F : K]i(σ1(u) + σ2(u) + · · ·+ σr(u)).

Theorem 5.7.2. If F is a finite dimensional Galois extension field of K and

AutKF = {σ1, . . . , σn},

then for any u ∈ F ,

NFK(u) = σ1(u)σ2(u) · · ·σn(u); and

T FK (u) = σ1(u) + σ2(u) + · · ·+ σn(u).

Theorem 5.7.3. Let F be a finite dimensional extension field of K. Then for allu, v ∈ F :

(i) NFK(u)NF

K(v) = NFK(uv) and T FK (u) + T FK (v) = T FK (u+ v);

(ii) if u ∈ K, then NFK(u) = u[F :K] and T FK (u) = [F : K]u;

(iii) NFK(u) and T FK (u) are elements of K. More precisely,

NFK(u) = ((−1)na0))

[F :K(u)] ∈ K and T FK (u) = −[F : K(u)]an−1 ∈ K,

63


where f = xn + an−1xn−1 + · · ·+ a0 ∈ K[x] is the irreducible polynomial of u;

(iv) if E is an intermediate field, then

NEK(NF

E (u)) = NFK(u) and TEK (T FE (u)) = T FK (u).

Definition 5.7.4. Let S be a nonempty set of automorphisms of a field F. S islinearly independent provided that for any a1, . . . , an ∈ F and σ1, . . . , σn ∈S(n ≥ 1):

a1σ1(u) + · · ·+ anσn(u) = 0 for all u ∈ F ⇒ ai = 0 for every i.

Lemma 5.7.5. If S is a set of distinct automorphisms of a field F, then S is linearlyindependent.

An extension field F of a field K is said to be cyclic (resp. abelian) if F isalgebraic and Galois over K and AutKF is a cyclic (resp. abelian) group. Ifin this situation AutKF is a finite cyclic group of order n, then F is said to be acyclic extension of degree n (and [F : K] = n by the fundamental theorem). Weknow that every finite dimensional extension of a finite field is a cyclic extension.

Theorem 5.7.6. Let F be a cyclic extension field of K of degree n, σ a generatorof AutKF and u ∈ F . Then

(i) TKF (u) = 0 if and only if u = v − σ(v) for some v ∈ F ;(ii) (Hilbert’s Theorem 90) NF

K(u) = 1K if and only if u = vσ(v)−1 for somenonzero v ∈ F .

Proposition 5.7.7. Let F be a cyclic extension field of K of degree n and supposen = mpt where 0 6= p = char K and (m, p) = 1. Then there is a chain ofintermediate fields F ⊃ E0 ⊃ E1 ⊃ . . . ⊃ Et−1 ⊃ Et = K such that F isa cyclic extension of E0 of degree m and for each 0 ≤ i ≤ t, Ei−1 is a cyclicextension of Ei of degree p.

Proposition 5.7.8. Let K be a field of characteristic p 6= 0. F is a cyclic extensionfield of K of degree p if and only if F is a splitting field over K of an irreduciblepolynomial of the form xp − x − a ∈ K[x]. In this case F = K(u) where u isany root of xp − x− a.

Corollary 5.7.9. If K is a field of characteristic p 6= 0 and xp − x − a ∈ K[x],then xp − x− a is either irreducible or splits in K[x].

Let K be a field and n a positive integer. An element ζ ∈ K is said to be annth root of unity provided ζn = 1K (that is, ζ is a root of xn − 1K ∈ K[x]).The set of all nth roots of unity in K forms a multiplicative subgroup of themultiplicative group of nonzero elements of K. ζ ∈ K is said to be a primitiventh root of unity provided ζ is an nth root of unity and ζ has order n in themultiplicative group of nth roots of unity. In particular, a primitive nth root ofunity generates the cyclic group of all nth roots of unity.

If char K = p and p|n, then n = pkm with (p,m) = 1 and m < n. Thusxn − 1K = (xm − 1K)p

k . Consequently the nth roots of unity in K coincide

64


with the mth roots of unity in K. Since m < n, there can be no primitive nthroot of unity in K. Conversely, if char K - n (in particular, if char K = 0), thennxn−1 6= 0, hence xn−1K is relatively prime to its derivative. Therefore xn−1khas n distinct roots in any splitting field F of xn − 1K over K. Thus the cyclicgroup of nth roots of unity in F has order n and F contains a primitive nth root ofunity. Note that if K does contain a primitive nth root of unity, then K containsn distinct roots of xn − 1K , hence F = K.

Lemma 5.7.10. Let n be a positive integer and K a field which contains a prim-itive nth root of unity ζ .

(i) If d|n, then ζn/d = η is a primitive dth root of unity in K.(ii) If d|n and u is a nonzero root of xd− a ∈ K[x], then xd− a has d distinct

roots, namely u, ηu, η2u, . . . , ηd−1u, where η ∈ K is a primitive dth root of unity.Furthermore K(u) is a splitting field of xd − a over K and is Galois over K

Theorem 5.7.11. Let n be a positive integer and K a field which contains aprimitive nth root of unity ζ . Then the following conditions on an extension fieldF of K are equivalent.

(i) F is cyclic of degree d, where d|n:(ii) F is a splitting field over K of a polynomial of the form xn− a ∈ K[x] (in

which case F = K(u), for any root u of xn − a);(iii) F is a splitting field over K of an irreducible polynomial of the form

xd − b ∈ K[x], where d|n (in which case F = K(v), for any root v of xd − b).

5.8 CYCLOTOMIC EXTENSIONS

A splitting field F over a field K of xn − 1K ∈ K[x] (where n ≥ 1) is calleda cyclotomic extension of order n. If char K = p 6= 0 and n = mpt with(p,m) = 1, then xn−1K = (xm−1)p

t so that a cyclotomic extension of order ncoincides with one of order m. Thus we shall assume that char K does not dividen (that is, char K = 0 or is relatively prime to n).

Theorem 5.8.1. Let n be a positive integer, K a field such that char K does notdivide n and F a cyclotomic extension of K of order n.

(i) F = K(ζ), where ζ ∈ F is a primitive nth root of unity.(ii) F is an abelian extension of dimension d, where d|φ(n)(φ the Euler funci-

ton); if n is prime, F is actually a cyclic extension.(iii)AutKF is isomorphic to a subgroup of order d of the multiplicative group

of units of Zn.

Let n be a positive integer, K a field such that char K does not divide n, and Fa cyclotomic extension of order n of K. The nth cyclotomic polynomial over Kis the monic polynomial gn(x) = (x− ζ1)(x− ζ2) · · · (x− ζr) where ζ1, . . . , ζrare all the distinct primitive nth roots of unity in F.

65


Proposition 5.8.2. Let n be a positive integer, K a field such that char K doesnot divide n and gn(x) the nth cyclotomic polynomial over K.

(i) xn − 1K =∏

d|n gd(x).(ii) The coefficients of gn(x) lie in the prime subfield P of K. If char K = 0

and P is identified with the field Q of rationals, then the coefficients are actuallyintegers.

(iii) Deg gn(x) = φ(n), where φ is the Euler function.

Proposition 5.8.3. Let F be a cyclotomic extension of order n of the field Q ofrational numbers and gn(x) the nth cyclotomic polynomial over Q. Then

(i) gn(x) is irreducible in Q[x].(ii) [F : Q] = φ(n), where φ is the Euler function.(iii) AutQF is isomorphic to the multiplicative group of units in the ring Zn.

5.9 RADICAL EXTENSIONS

Definition 5.9.1. An extension field F of a field K is a radical extension of K ifF = K(u1, . . . , un), some power of u1 lies in K and for each i ≥ 2, some powerof ui, lies in K(u1, . . . , ui−1).

Definition 5.9.2. Let K be a field and f ∈ K[x]. The equation f(x) = 0 issolvable by radicals if there exists a radical extension F of K and a splittingfield E of f over K such that F ⊃ E ⊃ K.

Lemma 5.9.3. If F is a radical extension field of K and N is a normal closure ofF over K, then N is a radical extension of K.

Theorem 5.9.4. If F is a radical extension field of K and E is an intermediatefield, then AutKE is a solvable group.

Corollary 5.9.5. Let K be a field and f ∈ K[x]. If the equation f(x) = 0 issolvable by radicals, then the Galois group of f is a solvable group.

Proposition 5.9.6. Let E be a finite dimensional Galois extension field of K withsolvable Galois group AutKE. Assume that char K does not divide [E : K].Then there exists a radical extension F of K such that F ⊃ E ⊃ K.

Corollary 5.9.7. Let K be a field and f ∈ K[x] a polynomial of degree n > 0,where char K does not divide n! (which is always true when char K = 0). Thenthe equation f(x) = 0 is solvable by radicals if and only if the Galois group of fis solvable.

66

Chapter 6

Commutative Rings and Modules

6.1 CHAIN CONDITIONS

Rings are not assumed to be commutative, nor to have identity elements.

Definition 6.1.1. A module A is said to satisfy the ascending chain condition(ACC) on submodules (or to be Noetherian) if for every chain A1 ⊂ A2 ⊂A3 ⊂ · · · of submodules of A, there is an integer n such that Ai = An for alli ≥ n.

A module B is said to satisfy the descending chain condition (DCC) onsubmodules (or to be Artinian) if for every chain B1 ⊃ B2 ⊃ B3 ⊃ · · · ofsubmodules of B, there is an integer m such that Ai = Am for all i ≥ m.

Definition 6.1.2. A ring R is left (resp. right) Noetherian if R satisfies theascending chain condition on left (resp. right) ideals. R is said to be Noetherianif R is both left and right Noetherian.

A ring R is left (resp. right) Artinian if R satisfies the descending chaincondition on left (resp. right) ideals. R is said to be Artinian if R is both left andright Artinian.

Definition 6.1.3. A module A is said to satisfy the maximum condition (resp.minimum condition) on submodules if every nonempty set of submodules ofA contains a maximal (resp. minimal) element (with respect to set theoreticinclusion).

Theorem 6.1.4. A module A satisfies the ascending (resp. descending) chaincondition on submodules if and only if A satisfies the maximal (resp. minimal)condition on submodules.

Theorem 6.1.5. Let 0 → Af−→ B

g−→ C → 0 be a short exact sequence ofmodules. Then B satisfies the ascending (resp. descending) chan condition onsubmodules if and only if A and C satisfy it.

Corollary 6.1.6. If A is a submodule of a module B, then B satisfies the ascend-ing (resp. descending) chain condition if and only if A and B/A satisfy it.

CHAPTER 6. COMMUTATIVE RINGS AND MODULES

Corollary 6.1.7. IfA1, . . . , An are modules, then the direct sumA1⊕A2⊕· · ·⊕An satisfies the ascending (resp. descending) chain condition on submodules ifand only if each Ai satisfy it.

Theorem 6.1.8. If R is a left Noetherian (resp. Artinian) ring with identity,then every finitely generated unitary left R-module A satifies the ascending (resp.descending) chain condition on submodules.

Theorem 6.1.9. A module A satisfies the ascending chain condition on submod-ules if and only if every submodule of A is finitely generated. In particular, acommutative ring R is Noetherian if and only if every ideal of R is finitely gener-ated.

Theorem 6.1.10. Any two normal series of a module A have refinements that areequivalent. Any two composition series of A are equivalent.

Theorem 6.1.11. A nonzero module A has a composition series if and only if Asatisfies both the ascending and descending chain conditions on submodules.

Corollary 6.1.12. If D is a division ring, then the ring MatnD of all n × nmatrices over D is both Artinian and Noetherian.

6.2 PRIME AND PRIMARY IDEALS

Theorem 6.2.1. An ideal P (6= R) in a commutative ring R is prime if and onlyif R\P is a multiplicative set.

The set of all prime ideals in a ring R is called the spectrum of R.

Theorem 6.2.2. If S is a multiplicative subset of a ring R which is disjoint froman ideal I of R, then there exists an ideal P which is maximal in the set of allideals of R disjoint from S and containing I. Furthermore any such ideal P isprime.

Theorem 6.2.3. Let K be a subring of a commutative ring R. If P1, . . . , Pn areprime ideals of R such that K ⊂ P1 ∪ P2 ∪ · · · ∪ Pn, then K ⊂ Pi for some i.

Proposition 6.2.4. If R is a commutative ring with identity and P is an idealwhich is maximal in the set of all ideals of R which are not finitely generated,then P is prime.

Definition 6.2.5. Let I be an ideal in a commutative ring R. The radical (ornilradical) of I, denoted Rad I, is the ideal ∩P , where the intersection is takenover all prime ideals P which contain I. If the set of prime ideals containing I isempty, then Rad I is defined to be R.

68


The radical of the zero ideal is sometimes called the nilradical or primeradical of the ring R.

Theorem 6.2.6. If I is an ideal in a commutative ring R, then Rad I = {r ∈ R :rn ∈ I for some n > 0}.

Theorem 6.2.7. If I, I1, I2, . . . , In are ideals in a commutative ring R, then:(i) Rad (Rad I) = Rad I;(ii) Rad (I1I2 · · · In) = Rad (

⋂nj=1 Ij) =

⋂nj=1RadIj;

(iii) Rad (Im) = Rad I.

Definition 6.2.8. An ideal Q(6= R) in a commutative ring R is primary if forany a, b ∈ R:

ab ∈ Q and a 6∈ Q⇒ bn ∈ Q for some n > 0.

In the rest of this section, all rings have identity.

Theorem 6.2.9. If Q is a primary ideal in a commutative ring R, then Rad Q isa prime ideal.

If Q is a primary ideal in a commutative ring R, then the radical P of Q iscalled the associated prime ideal of Q. One says that Q is a primary idealbelonging to the prime P or that Q is primary for P or that Q is P-primary.For a given primary ideal Q, the associated prime ideal Rad Q is clearly unique.However, a given prime ideal P may be the associated prime of several differentprimary ideals.

Theorem 6.2.10. Let Q and P be ideals in a commutative ring R. Then Q isprimary for P if and only if:

(i) Q ⊂ P ⊂ RadQ; and(ii) if ab ∈ Q and a 6∈ Q then b ∈ P .

Theorem 6.2.11. If Q1, Q2, . . . , Qn are primary ideals in a commutative ring R,all of which are primary for the prime ideal P, then

⋂ni−1Qi is also a primary

ideal belonging to P.

Definition 6.2.12. An ideal I in a commutative ring R has a primary decom-position if I = Q1 ∩ Q2 ∩ · · · ∩ Qn with each Qi primary. If no Qi containsQ1 ∩ · · · ∩ Qi−1 ∩ Qi+1 ∩ · · · ∩ Qn and the radicals of the Qi are all distinct,then the primary decomposition is said to be reduced (or irredundant).

Theorem 6.2.13. Let I be an ideal in a commutative ring R. If I has a primarydecomposition, then I has a reduced primary decomposition.

69


6.3 PRIMARY DECOMPOSITION

In this section all rings are commutative with identity and all modules are unitary.

Definition 6.3.1. Let R be a commutative ring with identity and B an R-module.A submodule A( 6= B) is primary provided that

r ∈ R, b 6∈ A and rb ∈ A⇒ rnB ⊂ A for some positive integer n.

Theorem 6.3.2. Let R be a commutative ring with identity and A a primarysubmodule of an R-module B. Then QA = {r ∈ R : rB ⊂ A} is a primary idealin R.

A primary submodule A of a module B is said to belong to a prime ideal Por to be a P-primary submodule of B if P = Rad QA = {r ∈ R : rnB ⊂ A forsome n > 0}. In particular, if J is a primary ideal, then QJ = J .

Definition 6.3.3. Let R be a commutative ring with identity and B an R-module.A submodule C of B has a primary decomposition if C = A1 ∩A2 ∩ · · · ∩An,with each Ai a Pi-primary submodule of B for some prime ideal Pi of R. If no Aicontains A1 ∩ · · · ∩ Ai−1 ∩ Ai+1 ∩ · · · ∩ An and if the ideals P1, . . . , Pn are alldistinct, then the primary decomposition is said to be reduced.

If C,Ai and Pi are as before and Pj 6⊂ Pi for all j 6= i, then Pi is said to bean isolated prime ideal of C. In other words, Pi is isolated if it is minimal in theset {P1, . . . , Pn}. If Pi is not isolated it is said to be embedded.

Theorem 6.3.4. Let R be a commutative ring with identity and B an R-module. Ifa submodule C of B has a primary decomposition, then C has a reduced primarydecomposition.

Theorem 6.3.5. Let R be a commutative ring with identity and B an R-module.Let C(6= B) be a submodule of B with two reduced primary decompositions,

A1 ∩ A2 ∩ · · · ∩ Ak = C = A′1 ∩ A′2 ∩ · · · ∩ A′s,

where Ai is Pi-primary and A′j is P ′j-primary. Then k = s and (after reorderingif necessary) Pi = P ′i for i = 1, 2, . . . , k. Furthermore, if Ai and A′i both arePi-primary and Pi is an isolated prime, then Ai = A′i.

Theorem 6.3.6. Let R be a commutative ring with identity and B an R-modulesatisfying the ascending chain condition on submodules. Then every submoduleA(6= B) has a reduced primary decomposition. In particular, every submoduleA(6= B) of a finitely generated module B over a commutative Noetherian ring Rand every ideal (6= R) of R has a reduced primary decomposition.

70


6.4 NOETHERIAN RINGS AND MODULES

Proposition 6.4.1. (Cohen) A commutative ring R with identity is Noetherian ifand only if every prime ideal of R is finitely generated.

If B is a module over a commutative ring R, then it is easy to see that I ={r ∈ R : rb = 0 for all b ∈ B} is an ideal of R. The ideal I is called theannihilator of B in R.

Lemma 6.4.2. Let B be a finitely generated module over a commutative ring Rwith identity and let I be the annihilator of B in R. Then B satisfies the ascend-ing (resp. descending) chain condition on submodules if and only if R/I is aNoetherian (resp. Artinian) ring.

Lemma 6.4.3. Let P be a prime ideal in a commutative ring R with identity. IfC is a P-primary submodule of the Noetherian R-module A, then there exists apositive integer m such that PmA ⊂ C.

Theorem 6.4.4. (Krull Intersection Theorem) Let R be a commutative ring withidentity, I an ideal of R and A a Noetherian R-module. If B =

⋂∞n=1 I

nA, thenIB = B.

Lemma 6.4.5. (Nakayama) If J is an ideal in a commutative ring R with identity,then the following conditions are equivalent.

(i) J is contained in every maximal ideal of R;(ii) 1r − j is a unit for every j ∈ J;(iii) If A is a finitely generated R-module such that JA = A, then A = 0;(iv) If B is a submodule of a finitely generated R-module A such that A =

JA+B, then A = B.

Proposition 6.4.6. Let J be an ideal in a commutative ring R with identity. ThenJ is contained in every maximal ideal of R if and only if for every R-module Asatisfying the ascending chain condition on submodules,

⋂∞n=1 J

nA = 0.

Corollary 6.4.7. If R is a Noetherian local ring with maximal ideal M, then⋂∞n=1M

n = 0.

Proposition 6.4.8. If R is a local ring, then every finitely generated projectiveR-module is free.

Theorem 6.4.9. (Hilbert Basis Theorem) If R is a commutative Noetherian ringwith identity, then so is R[x1, . . . , xn].

Proposition 6.4.10. If R is a commutative Noetherian ring with identity, then sois R [[x]].

71


6.5 RING EXTENSIONS

Definition 6.5.1. Let S be a commutative ring with identity and R a subring of Scontaining 1S . Then S is said to be an extension ring of R.

Definition 6.5.2. Let S be an extension ring of R and s ∈ S. If there existsa monic polynomial f(x) ∈ R[x] such that s is a root of f, then s is said tobe integral over R. If every element of S is integral over R, S is said to be anintegral extension of R.

Theorem 6.5.3. Let S be an extension ring of R and s ∈ S. Then the followingconditions are equivalent.

(i) s is integral over R;(ii) R[s] is a finitely generated R-module;(iii) there is a subring T of S containing 1S and R[s] which is finitely gener-

ated as an R-module;(iv) there is an R[s]-submodule B of S which is finitely generated as an R-

module and whose annihilator in R[s] is zero.

Corollary 6.5.4. If S is a ring extension of R and S is finitely generated as anR-module, then S is an integral extension of R.

Theorem 6.5.5. If S is an extension ring of R and s1, . . . , st ∈ S are integral overR, then R[s1, . . . , st] is a finitely generated R-module and an integral extensionring of R.

Theorem 6.5.6. If T is an integral extension ring of S and S is an integral exten-sion ring of R, then T is an integral extension ring of R.

Theorem 6.5.7. Let S be an extension ring of R and let R be the set of allelements of S that are integral over R. Then R is an integral extension ring of Rwhich contains every subring of S that is integral over R.

If S is an extension ring of R, then the ring R is called the integral closure ofR in S. If R = R, then R is said to be integrally closed in S. An integral domainR is said to be integrally closed provided R is integrally closed in its quotientfield.

Theorem 6.5.8. Let T be a multiplicative subset of an integral domain R suchthat 0 6∈ T . If R is integrally closed, then T−1R is an integrally closed integraldomain.

If S is an extension ring of R and I(6= S) is an ideal of S, it is easy to seethat I ∩ R 6= R and I ∩ R is an ideal of R. The ideal J = I ∩ R is calledthe contraction of I to R and I is said to lie over J. If Q is a prime ideal in anextension ring S of a ring R, then the contraction Q ∩ R of Q to R is a primeideal of R.

72


Theorem 6.5.9. (Lying-over Theorem) Let S be an integral extension ring of Rand P a prime ideal of R. Then there exists a prime ideal Q in S which lies overP (that is, Q ∩R = P ).

Theorem 6.5.10. (Going-up Theorem) Let S be an integral extension ring of Rand P1, P prime ideals in R such that P1 ⊂ P . If Q1 is a prime ideal of S lyingover P1, then there exists a prime ideal Q of S such that Q1 ⊂ Q and Q lies overP.

Theorem 6.5.11. Let S be an integral extension ring of R and P a prime idealin R. If Q and Q’ are prime ideals in S such that Q ⊂ Q′ and both Q and Q’ lieover P, then Q = Q′.

Theorem 6.5.12. Let S be an integral extension ring of R and let Q be a primeideal in S which lies over a prime ideal P in R. Then Q is maximal in S if andonly if P is maximal in R.

6.6 DEDEKIND DOMAINS

Definition 6.6.1. A Dedekind domain is an integral domain R in which everyideal (6= R) is the product of a finite number of prime ideals.

Definition 6.6.2. Let R be an integral domain with quotient field K. A fractionalideal of R is a nonzero R-submodule I of K such that aI ⊂ R for some nonzeroa ∈ R.

If I is a fractional ideal of a domain R and aI ⊂ R(0 6= a ∈ R), then aI isan ordinary ideal in R and the map I → aI given by x 7→ ax is an R-moduleisomorphism.

Theorem 6.6.3. If R is an integral domain with quotient field K, then the setof all fractional ideal of R forms a commutative monoid, with identity R andmultiplication given by IJ = {

∑ni=1 aibi : ai ∈ I; bi ∈ J ;n ∈ N∗}.

A fractional ideal I of an integral domain R is said to be invertible if IJ = Rfor some fractional ideal J of R. Thus the invertible fractional ideals are preciselythose that have inverses in the monoid of all fractional ideals.

(i) The inverse of an invertible fractional ideal I is unique and is I−1 = {a ∈K : aI ⊂ R}. Indeed for any fractional ideal I the set I−1 is easily seen tobe a fractional ideal such that I−1I = II−1 ⊂ R. If I is invertible and IJ =JI = R, then clearly J ⊂ I−1. Conversely, since I−1 and J are R-submodulesof K, I−1 = RI−1 = (JI)I−1 = J(II−1) ⊂ JR = RJ ⊂ J , hence J = I−1.(ii) If I, A,B are fractional ideals of R such that IA = IB and I is invertible,then A = RA = (I−1I)A = I−1(IB) = RB = B. (iii) If I is an ordinary idealin R, then R ⊂ I−1.

73


Lemma 6.6.4. Let I, I1, I2, . . . , In be ideals in an integral domain R.(i) The ideal I1I2 · · · In is invertible if and only if each Ij is invertible.(ii) If P1 · · ·Pm = I = Q1 · · ·Qn, where the Pi and Qj are prime ideals in R

and every Pi is invertible, then m = n and (after reindexing) Pi = Qi for eachi = 1, . . . ,m.

Every nonzero prime ideal in a principle ideal domain is both invertible andmaximal.

Theorem 6.6.5. If R is a Dedekind domain, then every nonzero prime ideal of Ris invertible and maximal.

Lemma 6.6.6. If I is a fractional ideal of an integral domain R with quotientfield K and f ∈ HomR(I, R), then for all a, b ∈ I : af(b) = bf(a).

Lemma 6.6.7. Every invertible fractional ideal of an integral domain R withquotient field K is a finitely generated R-module.

Theorem 6.6.8. Let R be an integral domain and I a fractional ideal of R. ThenI is invertible if and only if I is a projective R-module.

A discrete valuation ring is a principal ideal domain that has exactly onenonzero prime ideal; (the zero ideal is prime in any integral domain).

Lemma 6.6.9. If R is a Noetherian, integrally closed integral domain and R hasa unique nonzero prime ideal P, then R is a discrete valuation ring.

Theorem 6.6.10. The following conditions on an integral domain R are equiva-lent.

(i) R is a Dedekind domain;(ii) every proper ideal in R is uniquely a product of a finite number of prime

ideals;(iii) every nonzero ideal in R is invertible;(iv) every fractional ideal of R is invertible;(v) the set of all fractional ideals of R is a group under multiplication;(vi) every ideal in R is projective;(vii) every fractional ideal of R is projective;(viii) R is Noetherian, integrally closed and every nonzero prime ideal is max-

imal;(ix) R is Noetherian and for every nonzero prime ideal P of R, the localization

RP of R at P is a discrete valuation ring.

6.7 THE HILBERT NULLSTELLENSATZ

Classical algebraic geometry is the study of simultaneous solutions of poly-nomial equations: f(x1, x2, . . . , xn) = 0 (f ∈ S), where K is a field and

74


S ⊂ K[x1, . . . , xn]. A solution of this system is an n-tuple, (a1, . . . , an) ∈F n = F × F × · · ·F (n factors), where F is an algebraically closed extensionfield of K and f(a1, . . . , an) = 0 for all f ∈ S. Such a solution is called a zeroof S in F n. The set of all zeros of S is called the affine K-variety (or algebraicset) in F n defined by S and is denoted V (S). Thus

V (S) = {(a1, . . . , an) ∈ F n : f(a1, . . . , an) = 0 for all f ∈ S}

Note that if I is the ideal of K[x1, . . . , xn] generated by S, then V (I) = V (S).The assignment S 7→ V (S) defines a function from the set of all subsets ofK[x1, . . . , xn] to the set of all subets of F n. Conversely, define a function fromthe set of subsets of F n to the set of subsets of K[x1, . . . , xn] by Y 7→ J(Y ),where Y ⊂ FN and

J(Y ) = {f ∈ K[x1, . . . , xn] : f(a1, . . . , an) = 0 for all (a1, . . . , an) ∈ Y }.

Note that J(Y ) is actually an ideal of K[x1, . . . , xn]. The correspondence givenby V and J has the same formal properties as does the Galois correspondencebetween intermediate fields of an extension and subgroups of the Galois group.

Lemma 6.7.1. Let F be an algebraically closed extension field of K and let S, Tbe subsets of K[x1, . . . , xn] and X,Y subsets of F n. Then

(i) V (K[x1, . . . , xn]) = ∅; J(F n) = ∅; J(∅) = K[x1, . . . , xn];(ii) S ⊂ T ⇒ V (T ) ⊂ V (S) and X ⊂ Y ⇒ J(Y ) ⊂ J(X);(iii) S ⊂ J(V (S)) and Y ⊂ V (J(Y ));(iv) V (S) = V (J(V (S))) and J(Y ) = J(V (J(Y ))).

It is natural to ask which objects are closed under this correspondence, that iswhich S and Y satisfy J(V (S)) = S and V (J(Y )) = Y .

Theorem 6.7.2. (Noether Normalization Lemma) Let R be an integral domainwhich is a finitely generated extension ring of a field K and let r be the tran-scendence degree over K of the quotient field F of R. Then there exists an al-gebraically independent subset {t1, t2, . . . , tr} of R such that R is integral overK[t1, . . . , tr}.

Lemma 6.7.3. If F is an algebraically closed extension field of a field K and I isa proper ideal of K[x1, . . . , xn], then the affine variety V (I) defined by I in F n

is nonempty.

Proposition 6.7.4. (Hilbert Nullstellensatz) Let F be an algebraically closedextension field of a field K and I a proper ideal of K[x1, . . . , xn]. Let V (I) ={(a1, . . . , an) ∈ F n : g(a1, . . . , an) = 0 for all g ∈ I}. Then

Rad I = J(V (I)) = {f ∈ K[x1, . . . , xn] : f(a1, . . . , an) = 0 for all(a1, . . . , an) ∈ V (I)}. In other words, f(a1, . . . , an) = 0 for every zero (a1, . . . , an)of I in F n if and only if fm ∈ I for some m ≥ 1.

Let K be a field. Every polynomial f ∈ K[x1, . . . , xn] determines a functionF n → F by substitution: (a1, . . . , an) 7→ f(a1, . . . , an). If V = V (I) is an

75


affine variety contained in F n, the restriction of this function to V is called aregular function on V. The regular functions V → F form a ring Γ(V ) which isisomorphic to K[x1, . . . , xn]/J(V (I)). This ring is called the coordinate ringof V.

76

Chapter 7

Linear Algebra

7.1 MATRICES AND MAPS

Definition 7.1.1. Let R be a ring. An array of elements (aij) with aij ∈ R andn rows and m columns, is called an n×m matrix over R. Two n×m matrices(aij) and (bij) are equal if and only if aij = bij in R for all i, j. The elementsa11, a22, . . . are said to form the main diagonal. An n × n matrix with aij = 0for all i 6= j is called a diagonal matrix. If R has an identity element, theidentity matrix In is the n × n diagonal matrix with 1R in each entry on themain diagonal, that is, In = (δij). The n × m matrices with all entries 0 arecalled zero matrices. The set of all n × n matrices over R is denoted MatnR.The transpose of an n × m matrix A = (aij) is the m × n matrix At = (bij)such that bij = aji for all i, j.

Definition 7.1.2. If A = (aij) and B = (bij) are n×m matrices, then the sumA+B is defined to be the n×m matrix (cij), where cij = aij + bij . If A = (aij)is an m × n matrix and B = (bij) is an n × p matrix then the product AB isdefined to be the m× p matrix (cij) where cij =

∑nk=1 aikbkj . Multiplication is

not commutative in general. If A = (aij) is an n×m matrix, and r ∈ R, rA isthe n×m matrix (raij). Also, rIn is called a scalar matrix.

Theorem 7.1.3. If R is a ring, then the set of all n ×m matrices over R formsan R − R bimodule under addition, with the n ×m zero matrix as the additiveidentity. Multiplication of matrices, when defined, is associative and distributiveover addition. For each n > 0, MatnR is a ring. If R has an identity, so doesMatnR, namely the identity matrix.

Theorem 7.1.4. LetR be a ring with identity. LetE be a free leftR-module witha finite basis of n elements and F a free left R-module with a finite basis of melements. Let M be the left R-module of all n×m matrices over R. Then there

77

CHAPTER 7. LINEAR ALGEBRA

is an isomorphism of abelian group:

HomR(E,F ) ∼= M.

If R is commutative this is an isomorphism of left R-modules.

Definition 7.1.5. Let U = {u1, . . . , un} be a basis of E, and V = {v1, . . . , vm}a basis of F and f ∈ HomR(E,F ). There are elements rij of R such thatf(uk) = rk1v1 + rk2v2 + · · ·+ rkmvm. Define a map β : HomR(E,F )→M byf 7→ A, where A is the n ×m matrix (rij). The matrix of a homomorphismf ∈ HomR(E,F ) relative to the ordered bases U ofE and V of F is the n×mmatrix (rij) = β(f). Thus the ith row of the matrix of f consists of the coeffi-cients of f(ui) ∈ F relative to the ordered basis {v1, . . . , vm}. In the specialcase when E = F and U = V we refer to the matrix of the endomorphism frelative to the ordered basis U .

The image under f of an arbitrary element of E may be calculated from thematrix A = (rij) of f as follows. If u = x1u1 + · · ·+ xnun ∈ E (xi ∈ R), then

f(u) = f

(n∑i=1

xiui

)=

n∑i=1

xif(ui) =n∑i=1

xi

(m∑j=1

rijvi

)

=m∑j=1

(n∑i=1

xirij

)vj =

m∑j=1

yjvj,

where yj =∑n

i=1 xirij .

Theorem 7.1.6. Let R be a ring with identity and let E,F,G, be free left R-modules with finite ordered bases U, V,W respectively. If f ∈ HomR(E,F )had n × m matrix A (relative to bases U and V ) and g ∈ HomR(F,G) hasm×p matrix B (relative to bases V and W ), then gf ∈ HomR(E,G) has n×pmatrix AB (relative to bases U and W ).

Definition 7.1.7. Let R be a ring with identity and A ∈MatnR. A is said to beinvertible/nonsingular if there exists B ∈ MatnR such that AB = In = BA.The inverse matrix B if it exists, is unique, and denoted A−1. The productAC of two invertible matrices is invertible with (AC)−1 = C−1A−1. If A is aninvertible matrix over a commutative ring, then so is its transpose and (At)−1 =(A−1)t.

The matrix of a homomorphism of free R-modules depends on the choice ofordered bases in both domain and codomain. So, it will be helpful to know therelationship between matrices that represent the same map relative to differentpairs of ordered bases.

78


Lemma 7.1.8. Let R be a ring with identity and E,F free left R-modules withordered bases U, V respectively such that |U | = n = |V |. Let A ∈ MatnR.Then A is invertible if and only if A is the matrix of an isomorphism f : E → Frelative to U and V . In this case A−1 is the matrix of f−1 relative to V and U .

Theorem 7.1.9. Let R be a ring with identity. Let E and F be free left R-modules with finite ordered bases U and V respectively such that |U | = n, |V | =m. Let f ∈ HomR(E,F ) have n × m matrix A relative to U and V . Then fhas n×m matrix B relative to another pair of ordered bases of E and F if andonly if B = PAQ for some invertible matrices P and Q.

Corollary 7.1.10. Let R be a ring with identity and E a free left R-modulewith an ordered basis U of finite cardinality n. Let A be the n × n matrix off ∈ HomR(E,E) relative to U . Then f has n× n matrix B relative to anotherordered basis of E if and only if B = PAP−1 for some invertible matrix P .

Definition 7.1.11. Let R be a ring with identity. Two matrices A,B ∈ MatnRare said to be similar if there exists an invertible matrix P such that B =PAP−1. Two n × m matrices C,D are said to be equivalent if there existsinvertible matrices P and Q such that D = PCQ.

7.2 RANK AND EQUIVALENCE

Definition 7.2.1. Let f : E → F be a linear transformation of (left) vectorspaces over a division ring D. The rank of f is the dimension of Im f and thenullity of f is the dimension of Ker f .

Definition 7.2.2. If R is a ring with identity, then Rn will denote the free R-moduleR⊕· · ·⊕R (n summands). The standard (ordered) basis ofRn consistsof the elements ε1 = (1R, 0, . . . , 0), . . . , εn = (0, . . . , 0, 1R). The row space(resp. column space) of an n × m matrix A over a ring R with idenity is thesubmodule of the free left (resp. right) module Rm (resp. Rn) generated by therows (resp. columns) of A considered as elements of Rm (resp. Rn). If R is adivision ring, then the row rank (resp. column rank) of A is the dimension ofthe row (resp. column) space of A.

Theorem 7.2.3. Let f : E → F be a linear transformation of finite dimensionalleft (resp. right) vector spaces over a division ring D. If A is the matrix of frelative to some pair of ordered bases, then the rank of f is equal to the row(resp. column) rank of A.

79


Corollary 7.2.4. If A is an n×m matrix over a division ring D, then row rankA = column rank A.

Equivalent matrices over a division ring D will be characterized in terms of rankand in terms of the following matrices. If m,n are positive integers, then En,m

0

is defined to be the n×m zero matrix. For each r(1 ≤ r ≤ min(n,m)), En,mr is

defined to be the n×m matrix whose first r rows are the standard basis vectorsε1, . . . , εr of Dm and whose remaining rows are zero. So

En,mr =

(Ir 00 0

)A set of canonical forms for an equivalence relation R on a set X is a subset Cof X that consists of exactly one element from each equivalence class of R.

Theorem 7.2.5. Let M be the set of all n ×m matrices over a division ring Dand let A,B ∈M .

i) A is equivalent to En,mr if and only if rank A = r.

ii) A is equivalent to B if and only if rank A = rank B.iii) The matricesEn,m

r (r = 1, 2, . . . ,min(n,m)) constitute a set of canonicalforms for the relation of equivalence on M .

Definition 7.2.6. Let A be a matrix over a ring R with identity. Each of thefollowing is called an elementary row operation on A:

i) interchange two rows of A;ii) left multiply a row of A by a unit c ∈ R;iii) for r ∈ R and i 6= j, add r times row j to row i.

Elementary column operations on A are defined analogously (with left mul-tiplication in ii, iii replaced by right multiplication). An n × n elementary(transformation) matrix is a matrix that is obtained by performing exactly oneelementary row (or column) operation on the identity matrix In.

Theorem 7.2.7. Let A be an n ×m matrix over a ring R with identity and letEn (resp. Em) be the elementary matrix obtained by performing an elementaryrow (resp. column) operation T on In (resp. Im). Then EnA (resp. AEn) is thematrix obtained by performing the operation T on A.

Corollary 7.2.8. Every n × n elementary matrix E over a ring R with identityis invertible and its inverse is an elementary matrix.

Corollary 7.2.9. If B is the matrix obtained from an n×m matrix A over a ringR with identity by performing a finite sequence of elementary row and columnoperations, then B is equivalent to A.

80


Definition 7.2.10. We may define the rank of a homomorphism f : E → F offree R-modules to be the rank of Im f . Similarly the row rank of a matrix Aover R is defined to be the rank of the row space A.

Proposition 7.2.11. If A is an n × m matrix of rank r > 0 over a principal

ideal domain R, then A is equivalent to a matrix of the form(Lr 00 0

), where

L is an r× r diagonal matrix with nonzero diagonal entries d1, . . . , dr such thatd1 | d2 | · · · | dr. The ideals (d1), . . . , (dr) in R are uniquely determined by theequivalence class of A.

Proposition 7.2.12. The following conditions on an n× n matrix A over a divi-sion ring D are equivalent:

i) rank A = n;ii) A is equivalent to the identity matrix In;iii) A is invertible;iv) A is the product of elementary transformation matrices.

Definition 7.2.13. If (ci1, ci2, . . . , cim) is a nonzero row of a matrix (cij), thenits leading entry is cit where t is the first integer such that cit 6= 0. A matrixC = (cij) over a division ring D is said to be in reduced row echelon formprovided:

i) for some r ≥ 0 the first r rows of C are nonzero (row vectors) and all otherrows are zero;

ii) the leading entry of each nonzero row is 1D;iii) if cij = 1D is the leading entry of row i, then ckj = 0 for all k 6= i;iv) if c1j1 , c2j2 , . . . , crjr are the leading entries of rows 1, 2, . . . , r, then j1 <

j2 < · · · jr.

Proposition 7.2.14. i) If C is in reduced row echelon form, then rank C is thenumber of nonzero rows.

ii) If A is any matrix over D, then A may be changed to a matrix in reducedrow echelon form by a finite sequence of elementary row operations.

7.3 DETERMINANTS

Definition 7.3.1. Let B1, . . . , Bn and C be modules over a commutative ring Rwith identity. A function f : B1 × · · · × Bn → C is said to be R-multilinear iffor each i = 1, 2, . . . , n and all r, s ∈ R, bj ∈ Bj and b, b′ ∈ Bi:

f(b1, . . . , bi−1, rb+ sb′, bi+1, . . . , bn) = rf(b1, . . . , bi−1, b, bi+1, . . . , bn)+

sf(b1, . . . , bi−1, b′, bi+1, . . . , bn).

81


If C = R, then f is called an n-linear or R-multilinear form. If C = R andB1 = B2 = · · · = Bn = B, then f is called an R-multilinear form on B. The2-linear functions are usually called bilinear.

Definition 7.3.2. Let B and C be R-modules and f : Bn → C an R-multilinearfunction. Then f is said to be symmetric if

f(bσ1, . . . , bσn) = f(b1, . . . , bn) for every permutation σ ∈ Sn,

and skew-symmetric if

f(bσ1, . . . , bσn) = (sgn σ)f(b1, . . . , bn) for every σ ∈ Sn.

f is said to be alternating if

f(b1, . . . , bn) = 0 whenever bi = bj for some i 6= j.

Let B be the free R-module R ⊕ R and let d : B × B → R be defined by((a11, a12), (a21, a22)) 7→ a11a22 − a12a21. Then d is a skew-symmetric alter-nating bilinear form on B. If one thinks of the elements of B as rows of 2 × 2matrices over R, then d is simply the ordinary determinant function.

Theorem 7.3.3. If B and C are modules over a commutative ring R with iden-tity, then every alternatingR-multilinear function f : Bn → C is skew-symmetric.

Our chief interest is in alternating n-linear forms on the freeR-moduleRn. Sucha form is a function from (Rn)n = Rn ⊕ · · · ⊕Rn (n summands) to R.

Theorem 7.3.4. If R is a commutative ring with identity and r ∈ R, thenthere exists a unique alternating R-multilinear form f : (Rn)n → R such thatf(ε1, . . . , εn) = r, where {ε1, . . . , εn} is the standard basis of Rn.

Since the elements ofRn may be identified with 1×n row vectors, it is clear thatthere is an R-module isomorphism (Rn)n ∼= MatnR given by (X1, . . . , Xn) 7→A, where A is the matrix with rows X1, . . . , Xn. If {ε1, . . . , εn} is the standardbasis of Rn, then (ε1, . . . , εn) 7→ In under this isomorphism. Thus the multilin-ear form f may be thought of as a function whose n arguments are the rows ofn × n matrices. A multilinear form on MatnR is an R-multilinear form on(Rn)n whose arguments are the rows of n × n matrices considered as elementsof Rn.

82


Definition 7.3.5. Let R be a commutative ring with identity. The unique alter-nating R-multilinear form d : MatnR → R such that d(In) = 1R is called thedeterminant function on MatnR. The determinant of a matrix A ∈ MatnRis the element d(A) ∈ R and is denoted |A|.

Theorem 7.3.6. Let R be a commutative ring with identity and A,B ∈MatnR.i) Every alternating R-multilinear form f on MatnR is a unique scalar mul-

tiple of the determinant function d.ii) If A = (aij), then |A| =

∑σ∈Sn

(sgn σ)a1σ1a2σ2 · · · anσn.iii) |AB| = |A||B|.iv) If A is invertible in MatnR, then |A| is a unit in R.v) If A and B are similar, then |A| = |B|.vi) |At| = |A|.vii) If A = (aij) is triangular, then |A| = a11a22 · · · ann.viii) If B is obtained by interchanging two rows (columns) of A, then |B| =

−|A|. If B is obtained by multiplying one row (column) of A by r ∈ R, then|B| = r|A|. If B is obtained by adding a scalar multiple of row i (column i) torow j (column j) (i 6= j), then |B| = |A|.

If R is a field, we have a method for calculating A. Use elementary row andcolumn operations to change A into a diagonal matrix B = (bij), keepingtrack at each stage (via viii) of what happens to |A|. So, |B| = r|A|, hencer|A| = b11 · · · bnn. More generally the det of an n × n matrix A over any com-mutative ring with id may be calculated as follows. For each pair (i, j) let Aijbe the (n − 1) × (n − 1) matrix obtained by deleting row i and column j fromA. Then |Aij| ∈ R is called the minor of A = (aij) at position (i, j) and(−1)i+j|Aij| ∈ R is called the cofactor of aij .

Proposition 7.3.7. If A is an n × n matrix over a commutative ring R withidentity, then for each i = 1, 2, . . . , n,

|A| =n∑j=1

(−1)i+jaij|Aij|

and for each j = 1, 2, . . . , n,

|A| =n∑i=1

(−1)i+jaij|Aij|.

The first (second) formula for |A| is called the expansion of |A| along row i(column j).

Proposition 7.3.8. If A = (aij) is an n × n matrix over a commutative ring Rwith identity and Aa = (bij) is the n × n matrix with bij = (−1)i+j|Aji|, then

83


AAa = |A|In = AaA. Furthermore A is invertible in MatnR if and only if|A| is a unit in R, in which case A−1 = |A|−1Aa. The matrix Aa is called theclassical adjoint of A. Note that if R is a field, then |A| is a unit if and only if|A| 6= 0.

Corollary 7.3.9. (Cramer’s Rule) Let A = (aij) be the matrix of coefficients ofthe system of n linear equations in n unknowns

a11x1 + a12x2 + · · ·+ a1nxn = b1

an1x1 + an2x2 + · · ·+ annxn = bn

over a field K. If |A| 6= 0, then the system has a unique solution which is givenby:

xj = |A|−1(

n∑i=1

(−1)i+jbi|Aij|

)for j = 1, 2, . . . , n.

7.4 DECOMPOSITION OF A SINGLE LINEAR

TRANSFORMATION AND SIMILARITY

Let K be a field and φ : E → E a linear transformation of an n-dimensionalK-vector space E. HomK(E,E) is a ring with identity and a vector spaceover K with (kψ)(u) = kψ(u) (k ∈ K, u ∈ E,ψ ∈ HomK(E,E)). So iff =

∑kix

i is a polynomial in K[x], then f(φ) =∑kiφ

i is a well definedelement of HomK(E,E) (where φ0 = 1E as usual). Similarly the ring MatnKis also a vector space over K. If A ∈ MatnK, then f(A) =

∑kiA

i is a welldefined n× n matrix over K.

Theorem 7.4.1. Let E be an n-dimensional vector space over a field K, φ :E → E a linear transformation and A an n× n matrix over K.

i) There exists a unique monic polynomial of positive degree, qφ ∈ K[x], suchthat qφ(φ) = 0 and qφ | f for all f ∈ K[x] such that f(φ) = 0.

ii) There exists a unique monic polynomial of positive degree, qA ∈ K[x],such that qA(A) = 0 and qA | f for all f ∈ K[x] such that f(A) = 0.

iii) If A is the matrix of φ relative to some basis of E, then qA = qφ.

84


Definition 7.4.2. If K,E, and φ are as before, then the polynomial qφ (resp. qA)is called the minimal polynomial of the linear transformation φ (matrix A). Ingeneral, qφ is not irreducible. Now, φ induces a (left) K[x]-module structure onE as follows. If f ∈ K[x] and u ∈ E, then f(φ) ∈ HomK(E,E) and fu isdefined by fu = f(φ)(u). A K-subspace F of E is said to be invariant under φif φ(F ) ⊂ F . Clearly F is a φ-invariant K-subspace if and only if F is a K[x]-submodule of E. In particular, for any v ∈ E the subspace E(φ, v) spannedby the set {φi(v) : i ≥ 0} is φ-invariant. E(φ, v) is precisely the cyclic K[x]-submodule K[x]v generated by v. E(φ, v) is said to be a φ-cyclic subspace.

Theorem 7.4.3. Let φ : E → E be a linear transformation of an n-dimensionalvector space E over a field K.

i) There exist monic polynomials of positive degree q1, q2, . . . , qt ∈ K[x] andφ-cyclic subspaces E1, . . . , Et of E such that E = E1 ⊕ · · · ⊕ Et and q1 | q2 |· · · | qt. Furthermore qi is the minimal polynomial of φ|Ei : Ei → Ei. Thesequence (q1, . . . , qt) is uniquely determined by E and φ and qt is the minimalpolynomial of φ.

ii) There exist monic irreducible polynomials p1, . . . , ps ∈ K[x] and φ-cyclicsubspacesE11, . . . , E1k1 , E21, . . . , E2k2 , E31, . . . , Esks ofE such thatE =

∑si=1

∑kij=1Eij

and for each i there is a nonincreasing sequence of integers mi1 ≥ mi2 ≥ · · · ≥miki ≥ 0 such that pmij

i is the minimal polynomial of φ|Eij : Eij → Eij . Thefamily of polynomials {pmij

i : 1 ≤ i ≤ s; 1 ≤ j ≤ ki} is uniquely determined byE and φ and pm11

i pm212 · · · pms1

s is the minimal polynomial of φ.

Definition 7.4.4. The polynomials q1, . . . , qt in part i are called the invariantfactors of the linear transformation φ. The prime power polynomials pmij

i inpart ii are called the elementary divisors of φ.

Theorem 7.4.5. Let φ : E → E be a linear transformation of a finite dimen-sional vector space E over a field K. Then E is a φ-cyclic space and φ has min-imal polynomial q = xr +ar−1x

r−1 + · · ·+a0 ∈ K[x] if and only if dimKE = rand E has an ordered basis V relative to which the matrix of φ is

A =

0 1K 0 0 0 · · · 0 00 0 1K 0 0 · · · 0 00 0 0 1K 0 · · · 0 0· ·· ·· ·0 0 0 0 0 · · · 0 1K−a0 −a1 −a2 −a3 −a4 · · · −ar−2 −ar−1

In this case V = {v, φ(v), φ2(v), . . . , φr−1(v)} for some v ∈ E. This matrix iscalled the companion matrix of the monic polynomial q ∈ K[x].

Theorem 7.4.6. Let φ : E → E be a linear transformation of an n-dimensionalvector space E over a field K.

85


i) E has a basis relative to which the matrix of φ is the direct sum of thecompanion matrices of the invariant factors q1, . . . , qt ∈ K[x] of φ.

ii) E has a basis relative to which the matrix of φ is the direct sum of thecompanion matrices of the elementary divisors pm11

1 , . . . , pmskss ∈ K[x] of φ.

iii) If the minimal polynomial q of φ factors as q = (x−b1)r1(x−b2)r2 · · · (x−bd)

rd (bi ∈ K), which is always the case if K is algebraically closed, then everyelementary divisor of φ is of the form (x − bi)

j (j ≤ ri) and E has a basisrelative to which the matrix of φ is the direct sum of the elementary Jordanmatrices associated with the elementary divisors of φ.

Corollary 7.4.7. Let A be an n× n matrix over a field K.i) A is similar to a matrix D such that D is the direct sum of the companion

matrices of a unique family of polynomials q1, . . . , qt ∈ K[x] such that q1 | · · · |qt. The matrix D is uniquely determined.

ii) A is similar to a matrix M such that M is the direct sum of the companionmatrices of a unique family of prime power polynomials in K[x], where each isprime in K[x]. M is uniquely determined except for the order of the companionmatrices of the polynomials along its main diagonal.

iii) If K is algebraically closed, then A is similar to a matrix J such that Jis a direct sum of the elementary Jordan matrices associate with a unique familyof polynomials of the form (x − b)m (b ∈ K). J is uniquely determined exceptfor the order of the elementary Jordan matrices along its main diagonal.

Definition 7.4.8. The matrixD in part i is said to be in rational canonical form.The matrix M in part ii is said to be in primary rational canonical form andthe matrix J in iii is said to be in Jordan canonical form. The polynomials inpart i are called the invariant factors of the matrixA. The unique prime powerpolynomials in part ii are called the elementary divisors of the matrix A.

Corollary 7.4.9. Let φ : E → E be a linear transformation of an n-dimensionalvector space E over a field K.

i) If φ has matrix A ∈ MatnK relative to some basis, then the invariantfactors (resp. elementary divisors) of φ are the invariant factors (elementarydivisors) of A.

ii) Two matrices in MatnK are similar if and only if they have the sameinvariant factors (resp. elementary divisors).

Proposition 7.4.10. Let A be an n × n matrix over a field K. Then the matrixof polynomials xIn − A ∈ MatnK[x] is equivalent (over K[x]) to a diagonalmatrix D with nonzero diagonal entries f1, . . . , fn ∈ K[x] such that each fi ismonic and f1 | · · · | fn. Those polynomials fi which are not constants are theinvariant factors of A.

86


7.5 THE CHARACTERISTIC POLYNOMIAL, EIGENVECTORS

AND EIGENVALUES

Definition 7.5.1. IfA is an n×nmatrix over a commutative ringK with identity,then xIn −A is an n× n matrix over K[x] hence the determinant is an elementof K[x]. The characteristic polynomial of the matrix A is the polynomialpA = |xIn − A| ∈ K[x]. Clearly, pA is a monic polynomial of degree n. If B ∈MatnK is similar to A, then pB = pA. Let φ : E → E be an endomorphismof a free K-module E of finite rank n. The characteristic polynomial of theendomorphism φ, denoted pφ is defined to be pA, where A is any matrix of φrelative to some ordered basis.

Lemma 7.5.2. i) If A1, . . . , Ar are square matrices (of various sizes) over acommutative ring K with id and pi ∈ K[x] is the characteristic polynomial ofAi, then p1p2 · · · pr ∈ K[x] is the characteristic polynomial of the matrix directsum of A1, . . . , Ar.

ii) The companion matrix C of a monic polynomial f ∈ K[x] has character-istic polynomial f .

Theorem 7.5.3. Let φ : E → E be a linear transformation of an n-dimensionalvector space over a field K with characteristic polynomial pφ ∈ K[x], minimalpolynomial qφ ∈ K[x], and invariant factors q1, . . . , qt ∈ K[x].

i) The characteristic polynomial is the product of the invariant factors; thatis, pφ = q1q2 · · · qt = q1q2 · · · qt−1qφ.

ii) (Cayley-Hamilton) φ is root of its characteristic polynomial; that is, pφ(φ) =0.

iii) An irreducible polynomial in K[x] divides pφ if and only if it divides qφ.

Definition 7.5.4. Let φ : E → E be a linear transformation of a vector space Eover a field K. A nonzero vector u ∈ E is an eigenvector of φ if φ(u) = ku forsome k ∈ K. An element k ∈ K is an eigenvalue of φ if φ(u) = ku for somenonzero u ∈ E.

Theorem 7.5.5. Let φ : E → E be a linear transformation of a finite dimen-sional vector space E over a field K. Then the eigenvalues of φ are the roots ofK of the characteristic polynomial pφ of φ.

Definition 7.5.6. If k ∈ K is an eigenvalue of an endomorphism φ of aK-vectorspace E, then it is easy to see that C(φ, k) = {v ∈ E : φ(v) = kv} is a nonzerosubspace of E; C(φ, k) is called the eigenspace of k.

Theorem 7.5.7. Let φ : E → E be a linear transformation of a finite dimen-sional vector spaceE over a fieldK. Then φ has a diagonal matrixD relative tosome ordered basis of E if and only if the eigenvectors of φ span E. In this case

87


the diagonal entries of D are the eigenvalues of φ and each eigenvalue k ∈ Kappears on the diagonal dimKC(φ, k) times.

Definition 7.5.8. The eigenvalues and eigenvectors of an n×n matrix A overa field K are defined to be respectively the eigenvalues and eigenvectors of theunique linear transformation φ : Kn → Kn that has matrix A relative to thestandard basis.

Proposition 7.5.9. Let K be a commutative ring with id. Let φ be an endomor-phism of a free K-module of rank n and let A = (aij) ∈ MatnK be the matrixof φ relative to some ordered basis. If the characteristic polynomial of φ and Ais pφ = pA = xn + cn−1x

n−1 + · · ·+ c1x+ c0 ∈ K[x], then

(−1)nc0 = |A| and − cn−1 = a11 + a22 + · · ·+ ann.

Definition 7.5.10. Let K be a commutative ring with id. The trace of an n× nmatrix A = (aij) over K is a11 + · · · + ann ∈ K and is denoted Tr A. Thetrace of an endormorphism φ of a free K-module of rank n, denoted Trφ is TrA, where A is the matrix of φ relative to some ordered basis.

88

Abstract Algebra€¦ · Chapter 1 Groups 1.1SEMIGROUPS, MONOIDS AND GROUPS Deﬁnition 1.1.1. If G...

Documents

Transcript of Abstract Algebra€¦ · Chapter 1 Groups 1.1SEMIGROUPS, MONOIDS AND GROUPS Deﬁnition 1.1.1. If G...