0. Introduction - Math 407: Modern Algebra Icampbell/Math407Spr08/notes/00_Intro/slides.pdf · UMBC...

0. IntroductionMath 407: Modern Algebra I

Robert Campbell

UMBC

January 29, 2008

Robert Campbell (UMBC) 0. Introduction January 29, 2008 1 / 22

Outline

1 Math 407: Abstract Algebra

2 Sources

3 Cast of Characters

4 Background Material

5 Applications & Follow-Up


UMBC Course Description

The basic abstract algebraic structures (rings, integral domains, divisionrings, fields and Boolean algebra) will be introduced, and the fundamentalconcepts of number theory will be examined from an algebraic perspective.This will be done by examining the construction of the natural numbersfrom the Peano postulates, the construction of the integers from thenatural numbers, the rationals as the field of quotients of the integers, thereals as the ordered field completion of the rationals and the complexnumbers as the algebraic completion of the reals. The basic concepts ofnumber theory lead to modular arithmetic; ideals in rings; and to examplesof integral domains, division rings and fields as quotient rings. The conceptof primes yields the algebraic concepts of unique factorization domains,Euclidean rings, and prime and maximal ideals of rings. Examples ofsymmetries in number theory and geometry lead to the concept of groupswhose fundamental properties and applications will be explored.


Algebra

Def: Algebra is a branch of mathematics that

utilizes symbols, as letters, to represent specific numbers, values ofvectors. (Webster)

concerns the study of structure, relation and quantity. (Wikipedia)

From the title “Hisab al-jabr w’al-muqabala, Kitab al-Jabrwa-l-Muqabala” (The Compendious Book on Calculation byCompletion and Balancing) by Al-Khwarizmi (circa 820 AD)

Algebra generally refers to polynomial equations (aka algebraic equations)


Abstract

Def: Abstract

(adj) Thought of apart from concrete realities or specific objects.

(vt) To summarize

cow, cow, cow, dog, dog, dog −→ 3

Loss, debt −→ (−3)

Hypotenuse of isosceles right triangle −→√

2 (irrational)

x2 + 1 = 0 −→ i =√−1


Outline


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Number Theory

aka Algebra over the integers

Solve 6x + 15y = 9 (Linear Diophantine equation)

Solve x2 + y2 = z2 (Pythagorean triples)

Solve x3 + 3y2 = 5 (Elliptic curve)

Solve x3 + y3 = z3 (subcase of Fermat’s Last Theorem)


Geometry

The sides of an isosceles right triangle are incommensurable (√

2 isirrational) [Pythagorus, ca 500 BC]

Angles cannot be trisected (with compass and ruler)

A right triangle whose sides are integers cannot have area which is asquare or twice a square [Conj: Fermat, 165?, unproven]


Algebra

Solutions to quadratic equations

Solutions to cubic and quartic equations

Can solutions to x5 + ax4 + 1 = 0 be expressed with just roots?

Solutions to systems of linear equations

Solutions to systems of polynomial equations


Symmetries

Geometries: Groups of Isometries (distance preservingtransformations)

Symmetry Groups

Geometric FiguresTesselation & Crystallographic Groups

Topology


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The Zoo

Groups

Rings

Fields

Vector Spaces & Modules

Algebras


Groups

Def: A group is a set with a “multiplication” operation, an identityelement (“multiplication” by it has no effect) and inverses.

Integers with Addition: Z+

Modular Integers with Multiplication: Z∗n

Matrix Groups with Multiplication: GLn(R), SL2(Z), etc

Permutation Groups:

Geometric Symmetries: Tesselations, Polygons, Polyhedra


Rings

Def: A ring is a set with a “multiplication” operation and a commutative“addition” operation, an element which acts like “0” and another whichacts like “1”, and additive inverses.

Z, Q, R and CSquare Matrices: Mn(R), Mn(Z), etc

Polynomials: Z[x ], C[x ], Q[x , y ], etc

Modular Integers: Zn

Algebraic Integers: Z[√−1], Z[

√3], etc

Real Division Rings: R ⊂ C =< 1, i |i2 = −1 >⊂ H =< 1, i , j , k|i2 =j2 = k2 = ijk = −1 >


Fields

Def: A field is a set with a commutative “multiplication” operation and acommutative “addition” operation, an element which acts like “0” andanother which acts like “1”, additive and multiplicative inverses (exceptfor “0”).

Q, R and CNumber Fields: Q[

√−1], Q[

√3], etc

Finite Fields: Zp, GF (pn) := Zp[x ]/ < p(x) >


Vector Spaces & Modules

Def:

A vector space V over a field F is a set with commutative additionand scalar multiplication by elements of the field.

A module M over a ring R is a set with commutative addition andscalar multiplication by elements of the ring.

Vector Space: Rn, Cn, Mn(R), R[x ], etc

Module: Zn, Zk1 ⊕ Zk2 ⊕ · · · , etc


Algebras

Def: An algebra is a ring which is also a vector space.

Square Matrices: Mn(R), etc

Polynomials: C[x ], Q[x , y ], etc

Real Division Algebras: R ⊂ C =< 1, i |i2 = −1 >⊂ H =<1, i , j , k|i2 = j2 = k2 = ijk = −1 >


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Linear Algebra

Matrix multiplication

Non-commutative multiplication (AB 6= BA)

Zero divisors (e.g.

(0 10 0

) (1 00 0

)=

(0 00 0

))

Nilpotent elements (e.g. if N =

(0 10 0

)then N2 = 0)

Idempotent elements (e.g. if A =

(1 00 0

)then A2 = A, but A 6= 1)

Trace and Det of a linear transformation


Analysis

Logic

Proofs

Set Theory

C and R - Construction and algebraic closure.

Polynomial and rational functions


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Further Topics

Abstract Algebra II (Math 408)

more Rings & Fieldsmore Group TheoryGalois Theory (combining Group & Field Theory)

Number Theory (Math 413)

Algebraic Number Theory

Algebraic Geometry

Algebraic Topology

Differential Geometry


0. Introduction - Math 407: Modern Algebra Icampbell/Math407Spr08/notes/00_Intro/slides.pdf · UMBC...

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