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R C
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R 1
R[X] = n
i=0
aiXi : n N a1, . . . , an R .R[X1, . . . , X n]
f(X) =n
i=0 aiXi R[X] R R, b ni=0 aibi =
f(b) b f(b) f(X)R = Fp =
ZpZ f(X) = X
p g(X) = X f(b) = g(b)b Fp f = g
R = K K[X]f, g K[X] g = 0 q, r K[X]
f = qg + rdeg r < deg g
g = X a f(X) = (X a)q(X) + f(a)r deg r < 1 = deg(X a)
K[X]f, g K[X]
gcd(f, g) = pf + qg p, q K[X]K 0 = f K[X] f deg f
K
f c Kf(X) = (X c)q(X) deg q = deg f 1 b
f (b c)q(b) = 0 c = b b q K|{ f}| 1 + |{ q}|
1 + deg q= deg f
f
X2 1 R[X] R = Z/8Z 1, 3R
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n 1 Sn n
{1, . . . , n
} R X1, . . . , X nf(X1, . . . , X n) R[X1, . . . , X n]
Sn f(X(1), . . . , X (n)) = f(X1, . . . , X n).f, g f + g f g
R[X1, . . . , X n]R
pr = Xr1 + + Xrn r 0
SnR[X1, . . . , X n] : f f(X(1), . . . , X (n)) = f(f) = f
R[X1, . . . , X n] Sn
n
i=1(T + Xi) = (T + X1)(T + X2) (T + Xn)
= Tn + (X1 + + Xn)Tn1 + + X1 Xn
sr = sr,n Tnr
sr =
1i1
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: R[Y1, . . . , Y n] R[X1, . . . , X n], G(Y1, . . . , Y n) G(s1, . . . , sn).
ker = {0} R[Y1, . . . , Y n] X1, . . . , X n
I = (i1, . . . , in) ik 0 k = 1, . . . , n XI =Xi11 Xinn
R[X1, . . . , X n] R XIi1 + + in
f R[X1, . . . , X n]f = f0 + f1 +
+ fd d fk 0
k {X1, . . . , X n}f f0, f1, . . . , f d
fd
{XI} XI > XJi1 > j1 p > 1 i1 = j1, . . . , ip1 = jp1 ip > jp
{XI}I, J XI > XJ, XI < XJ, XI = XJ
f dXI f c R
i1
i2
in ip < ip+1Xp Xp+1 XI
I = (i1, . . . , ip1, ip+1, ip, ip+2, . . . ) XI > XI
XI = Xi1i21 (X1X2)
i2i3 (X1 Xn1)in1in(X1 Xn)in .
g = si1i21 si2i32 sin1inn1 sinn d
XI srX1X2 Xr
h = fcg h h dXI
f {sr}G R[Y1, . . . , Y n] G(s1,n, . . . , sn,n) = 0
n G = 0 n = 1Ykn G k > 0 H = G/Y
kn
(H(s1,n, . . . , sn,n) = 0 Yn Yn GXn = 0
sr,n(X1, . . . , X n1, 0) =
sr,n1 r < n0 r = n
G(s1,n
1, . . . , sn
1,n
1, 0) = 0 G(Y1, . . . , Y n
1, 0) =
0 Yn | G
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i=j X
2i Xj X
21 X2 =
X1(X1X2)
s1s2 = ij
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= (1)n(n1)/2i=j
(Xi Xj)
n = 2
2(X1, X2) = (X1 X2)2 = (X1 + X2)2 4X1X2= s21 4s2.
f(T) =n
i=1
(T i)
= Tn
c1T
n1 +
+ (
1)ncn
cr = sr(1, . . . , n) f
Disc(f) = 2(1, . . . , n)
=i
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1 1 = 0
K
n Z n = 0 n.1K = 0K
n.1K =
1K + + 1K n > 0(n).1K n < 0
K 0 n Z n = 0 n.1K = 0K K
p p.1K = 0K Kp
K K
char K = 0 m.1Kn.1K
| n = 0
= Q
char K = p > 0
{m.1K | 0 m p 1} = Z/pZ = FpK L K
L L K L/K
Q R C K K(X)fg : f, g K[X], g = 0
K
i : K LK {0}
L K K L
C = {(x, y) : x, y R} +, i = (0, 1) R ={(x, 0) : x R}
L/K L KL K
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L K L/K[L : K] = dimK L L/K L/K
[L : K] =
[L : K] = n N L = Kn KC/R 2 {1, i} C
RK [K(X) : K] = 1, X , X 2, . . .
KR/Q
2 3
K p > 0 |K| = pn
n 1 n = [K : Fp]
n = [K : Fp] = dimFp K K K= Fnp
Fp |K| = pn
L/K n VL dimK V = n dimL V V K
L
dimL V = d 1 dk < n x G xdk = 1Xdk 1 n F F
R p p.1R = 0Rp : R
R p(x) = x
p
R
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p(1) = 1 p(xy) = p(x)p(y) p(x + y) = p(x) +p(y)
p(x + y) = (x + y)p
= xp +
p1r=1
p
r
xrypr + yp
= xp + yp
= p(x) + p(y)
1 r < p pr 0 (mod p)p
a Z ap a (mod p)
a (a + 1)p ap + 1
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L/K x L
K[x] =
n
i=0
aixi : n 0 a1, . . . , an K
L,
K(x) =y
z: y, z K[x] z = 0
L.
K[x] L K(x) L K[x]L K x K(x) L K
x K[x] K(x) x K
C/Q Q[i] = {a + bi : a, b Q} Q[i] = Q(i)
(a + bi)1 = aa2 + b2
ba2 + b2
i
a + bi = 0x K
f(X) K[X] f(x) = 0 x K
x K m(X) K[X]m(x) = 0 m(X) m = f g deg f, deg g < deg m
f(x) g(x) 0 f(X) K[X] f(x) = 0 m | ff(X) = q(X)m(X) + g(X)
deg g < deg m g(x) = 0 g(X)m(X)
x : K[X] Lf(X) f(x)
x K ker(x) = {0}ker(x) K[X]ker(x) = (m) m(X) K[X]
x K[x] L
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f(X), g(X) f(x) = 0g(y) = 0 h(X) h(x + y) = 0
K = Q m, n
Z x =
m, y =
n, f(X) = X2
m, g(X) = X2
n
z = x + y = m + n z2 = m + 2mn + n (z2 m n)2 = 4mnx + y
x X3 + X+ 3 y X4 + 2X3 + 2f(X) f(x + y) = 0
a, b K = a = b = +
2 = ( + )2 = a + b + 2
4 = (a + b)2 + 4(a + b) + 422
= (a
2
+ 6ab + b
2
) + 4(a + b)
4 2(a + b)2 = (a b)2
f(X) = X4 2(a + b)X2 + (a b)2degK = m degK = n
ij 0 i < m, 0 j < nK[, ] K[, ]
1, , 2, . . . , mn
[K() : K] | 4K = Q m,n,mn [Q(
m +
n) : Q] = 4
L/K x L K
K(x)/K x K K(x)/K
M/L/K M/KM/L L/K
L/K x
L K(x)/K xK x L L/K
K(x)/K x K K(x)/KK(x)/K
M/K x M K L/Kx M L K M/L
M/L/K L/K x Mx L x K
xL f(x) = 0 f(X) = Xd + a1Xd1 + + ad L[X]
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L0 = K(a1, . . . , ad) L/K a1, . . . , ad KL0/K f L0[X] x L0
[L0(x) : L0] < [L0(x) : K] < [K(x) : K] < xK
Q = {x C : x Q}x, y Q x y,xy,x1 Q Q C Q
Q Q Q Q Q( n2) Xn 2Q n
2 [Q( n
2) : Q] = n
n N [Q : Q] = L = Q( 3
2, 4
5) [L : Q] = 12 [Q( 3
2) : Q] = 3
X3 2 3 | [L : Q] [Q( 45) : Q] = 4 X4 54 | [L : Q] 12 | [L : Q]
X4
5 4
5 Q
45 Q( 32) [L : Q] | 12 = e2i/p + e2i/p p degQ
= e2i/p p = 1
Xp 1X 1 = 1 + X+ + X
p1 = f(X)
f(X) Q [Q() : Q] = p 1 Q() = + 1Q Q() Q() degQ | p 1
= 2 + 1 X2 X + 1 Q()[X] 2
Q() Q()
R Q()
[Q() : Q()] = 2 [Q() : Q] = (p 1)/2
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R C
x R x C QQ
x f(x) = 0f(X) = cdX
d +cd1Xd1 + +c0 c0, . . . , cd Z cd > 0 gcd(c0, . . . , cd) =1 f(X) f(X)
x
H(x) = d + |c0| + + |cd| NC N f(X) d +di=0|ci| C
x Q H(x) C Q
n=1
1
22n2
19 e 20 xy
x, y x = 0 y Q e
= (1)i
xy11 xymm xi, yixi = 0 1, y1, . . . , ym Q e
x =
n=11
22n2
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R C
k(n) = 2n2
f(X) = Z[X] d > 0f(x) = 0 xn =
nm=1
12k(m)
|x xn| =
m=n+1
1
2k(m)
j=0
1
2k(n+1)+j =
2
2k(n+1)
f(xn) n f(xn) = 0 f(X)f(xn) xn 2
k(n)
f(xn) 2dk(n) n |f(xn)| 12dk(n)
f(x) = 0 f(X) = (X x)g(X) g(X) R[X]
|g(xn)| = |f(xn)||xn x|
1
2dk(n)
2k(n+1)
2= 2k(n+1)dk(n)1
k(n + 1) dk(n) = 2(n+1)2 d2n2 = 2n2(22n+1 d) n |g(xn)| n
limn g(xn) = g
lim
n xn
= g(x) =
k(n)k(n + 1) dk(n) n k(n) = n!
(n + 1)! dn! = (n + 1 d)n! n=1 12n!
(x1, y1), . . . , (xm, ym) R2P1, P2, Q1, Q2 Pi = Qi
P1Q1 P2Q2P1, P2, Q1, Q2 Pi = Qi
Pi QiP1 = P2
P1, P2, Q1, Q2 Pi = QiP1Q1 P2 Q2
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= QRP
P
circ(P, Q)circ(Q, Q)circ(Q, Q)
S, TST
p circ(P, Q) Q
circ(Q, P)circ(Q, P) S
P S P
P P
R, S = P Q P
R
circ(R, S) TP R
P R T P
P P
RS
(0, 0), (0, 1)
(x, y) R2 (x1, y1), . . . , (xm, ym)
x R (x, 0) (0, 0) (1, 0)P = (a, b) R2 a, b R
Pa, b a x b
y x (a, 0)y (0, b)
(a, b)
R
a, b a+b ab a 1aa + b a
r, s r, srs =
rs
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R C
a > 0
a
a+12
GGGGGGGGGGGGG
oooooo
oooooo
oooooo
oooooo
oo
a
a 1
aK R K n 0
R
Q = F0 F1 Fn Rai Fi 1 i n
K FnFi = Fi1(ai)a2i Fi1
[Fi : Fi1] {1, 2} [Fi :Fi1] = 2 Fi = Fi1(
b) b Fi
K K/Q [K : Q] 2
x R Q(x)
x y xy xy
x x, y
k (x, y) k(0, 0), (0, 1) Q(x, y)
Q = F0 F1 Fn R(0, 0), (0, 1) k
Fn
(x, y) k + 1x, y Fn
x =a be y = c de a,b,c,d,e
x, y
Fn(
e)
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x R x Q2
3
2 degQ3
2 = 3 2
1
2/3
2/32/9 cos2/9
sin2/9
cos3 = 4cos3 3cos cos2/9 8X3 6X + 12(cos2/9 1) X3 + 6X2 + 9X+ 3 [Q (cos2/9) :Q] = 3 2
p pp 1 2
p 2/pcos2/p degQ 2/p =
(p 1)/2
n n = 2p
1pl
p1, . . . , pl 22k + 1
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f(X) K[X] dL/K d f(X)
(f) K[X] f(X)
Lf = K[X]/(f)
K K[X] K Lfx = X + (f) Lf f(x) = f(X) + (f) = 0 Lf x f Lf
Lf K f
L/K, M/K L MK K K
L/K L/K : K K : L L
x K (x) = (x)
L L K =
|K
L/K f(X) K[X]x L f(X) K
: Lf = K[X]/(f) L
X+ (f) xK Lf L
f(X) L deg f
K : Lf
L : K[X] L (a) = a
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a K (f) = 0 ker = (f)
Lf = K[X]/(f) // L
K[X]
OO
=
77ooooooooooo
(X)
aiXi
=
(ai)(X)i =
ai(X)
i
ai K (f) = 0 f((X)) = 0
: K[X] L(f) = 0
//
f(X) L 1 // (X)
L = K(x) K f(X)x K K : Lf
L X+ (f)x
K [Lf :K] = [L : K] = deg f
x, y K x, yK K : K(x)
K(y) xy x, y K
{f(X) K[X] : f(x) = 0} = {f(X) K[X] : f(y) = 0}
f(X)x, y
K(y) Lf = K[X]/(f)oo // K(x)
y X+ (f)1oo 1 // x
x = i y = i X2 + 1 Q : Q(i) Q(i) (i) = i
x = 3
2 y = e2i/3 3
2 X3 2 x, y
Q Q( 3
2) Q(e2i/3 32)
3
2 e2i/3 3
2
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f(X) K[X] : K Lf L[X] f
x L f : Lf = K[X]/(f) L : X+ (f) x : Lf L
f L
K f(X) K[X] L/Kf(X)
f(X) L[X]x1, . . . , xn L f(X) L L = K(x1, . . . , xn)
f(X)
K = Q
f(X) = X2 + 1 f(X) = (X + i)(X i) Q(i)/Qf(X) Q(i) f(X)
f(X) = X3 2 L = Q 32, e2i/3 f(X) Q[Q( 3
2 : Q] = 3 = e2i/3 Q( 32) R 2 + + 1 = 0
[L : Q( 3
2)] = 2 [L : Q] = 6 Lf(X) 3
f(X) = (X5 1)/(X 1) = X4 + X3 + X2 + X + 1
f(Y + 1) =(Y + 1)5 1
Y= Y4 + 5Y3 + 10Y2 + 10Y + 5
= e2i/5 f(X)f(X) C 2, 3, 4 L = Q() f(X) Q
f(X) K[X]f(X) K
f(X) K
d
1 d
d = 1 deg f = d + 1 g(X)f(X) deg g 1 K1 = K[X]/(g) x1 = X+ (g) K1
g(x1) = 0 f(x1) = 0 f(X) = (X x1)f1(X) deg f1 = df1 K1[X] f1(X) K1 L = K1(x2, . . . , xn)
x2, . . . , xn f1(X) L f(X)L[X] L = K(x1, x2, . . . , xn) x1, x2, . . . , xn f(X) L L
f(X) K
K Cf(X) K[X] L/K
f(X) : K
M fM[X]
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: L M [L : K] f(X)
LM f (K)
f(X) K
n = [L : K] n = 1 f(X)K n > 1 L = K x1, . . . , xm
f(X) L x1 KK1 = K(x1) d = degK(x1) = [K1 : K] > 1 g(X) x1K f(X) f(X) Lg(X) : K M 1 : K1 M
1 d g(X)1 [L : K1] 1 : L M
[L : K1][K1 : K] = [L : K] f(X)L 1 [L : K1]
[L : K1]d = [L : K]
: L M f M {(xi)}M f (K) M = (K)((x1), . . . , (xm)) = (L)
K L/K, M/KN/M K : L N
K
L = K(x1, . . . , xn) {x1, . . . , xn} L f(X) K[X]x1, . . . , xn L
f(X) L N f(X) M
L
// N
L
!!fff
f M
}}{{{{
K
L f(X) K L = K(x1, . . . , xn)x1, . . . , xn f(X) f(X) N[X]
K : L NL L
L L/K
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R C f(X) X = a f(a) =f(a) = 0
R f(X) R[X] f(X) = di=0 aiXif(X) =
di=0 iaiX
i1
(f + g) = f + g (f g) = f g + fg (fn) = nffn1
f(X) K[X] L/K x L f(X)x f(X) (X x)2 f(X) f(x) = 0
f(X) = (Xx)g(X) g(X) L[X] x f(X)g(x) = 0 f(X) = g(X) + (X x)g(X) f(x) = g(x)
K p > 0 b K bp K
f(X) = Xp b K[X] L/K f(X) a Lf(X) L f(X) = pXp1 = 0 X = a
ap = b L[X] f(X) = Xp ap = (X a)p X = af(X) f(X) K f(X) = g(X)h(X)
g(X), h(X) K[X] L[X] g(X) = (X a)m 0 < m < pg(X) = Xm maXm1 + K[X] ma K m 0 (mod p)
a K b p(K, b) K = Fp(X) b = X
f(X)f(X)
f(X) gcd(f(X), f(X)) = 1
f(X) f(X), f(X)
f, g K[X] gcd(f, f) K[X] L[X]L/K
f K[X] ff
= 0char K = 0 f K[X]
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char K = p > 0 f K[X]f(X) = g(Xp) g K[X]
f gcd(f, f)|
f gcd(f, f) {
1, f
}f = 0 gcd(f, f) = f f f = 0 gcd(f, f) | fdeg gcd(f, f) deg f < deg f gcd(f, f) = 1 f
f(X) =d
i=0 aiXi f = 0 iai = 0 1 i d
char K = 0 ai = 0 i 1 fchar K = p > 0 ai = 0 p i f(X) = g(X
p)g(X) =
0jd/p apjX
i
x K x K
L/K L/K x LK
x K x K X xchar K = 0 x K
0
x K f K[X]L f x
K deg f K K(x) LL = K(x1, . . . , xr, y)
K xi K Kc1, . . . , cr K L = K(z) z = y + c1x1 + + crxr
r r = 1 L = K(x, y)x K y K f, g x, y
M/L f g x = x1, x2, . . . , xmg M x f(X) =
ni=1(X xi)
c K mn yj + cxi Kc z = y + ckf(X) K[X] g(z cX) K(z)[X] x
z cx = y
f xi z cxi = yj jyj + cxi = z = y1 + cx1 i = 1 xi = x cf(X) g(z cX) X x f
f(X), g(z cX) K(z)[X] K(z)[X]x K(z) y = z cx K(z) K(z) = L
L/K L = K(x) x LL/K
K L = K(x1, . . . , xr) x1, . . . , xr KK L L
x L = K(x)
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Kf
K[X] K[X]
C Q
KL/K x L K x KL/K L = K
= x K xK x K
= f K[X] L f KL = K f K[X]= L/K x L K
L = K
L/KK[X] L L
L K
L/L L = Lx L L/L L/K x K f K[X]
f L[X] x LL = L
K = Q L = Q Q
L/K : K MK
: L M
L = K(x) f x Ky M f M[X] K(x)
M x
y
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S x S x x x, y S x y y x = x = y x,y ,z S x y y z = x z
S T S
x, y T x y y x.
S z S z xz = x T S z Sx T x z S
R
L/KS = {(L1, 1)} K L1 L 1 : L1 M (L1, 1) (L2, 2) L1 L2 2|L1 = 1 2(x) = 1(x) x L1
S
{(Li, i) : i I} S I L =
iI Lix Li y Lj Li Lj x, y Lj
x y, xy , xy Lj : L M (x) = i(x) i Ix Li x Li x Lj Li Lj
j|Li = i i(x) = j(x) i I (Li, i) (L, )(L, )
S S
(L, ) (L L x L \ LL : L M : L(x) M
(L, ) (L(x), ) L = L(x) (L, ) L = L =
K K K : K
K K K : K
K
K K
K K
K[X]
K K[X] = {f1, f2, . . . }K = K0 K1 Kn
fn Kn1 K =
nN Kn K[X] KK K
f K[X] Mff K Mf = K(xf,1, . . . , xf,d(f)) {xf,i}d(f)i=1
f Mf Mf= Rf/If Rf = K[Xf,1, . . . , X
f,d(f)] If
Rf Mf, Xf,i xf,i
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S K[X]
RS = K[{Xf,i}fS,1id(f)]
IS Rs {If}fSRS =
T RT IS =
T IT
T SIS = RS
S M fS =
fS f fK f: Mf M K Rf M, Xf,i f(xf,i)
RS M IfIS IS = RS
IS = RS 1
IS 1
IT T
S IT = RT
S K[X]J RS IS
RS/IS K = RS/JK K K
f Rf RS Rf RS/J = KIf
K Mf = Rf/If K f K KMf K K K
K : K
K K : K KK (K) K K/(K) K (K) = K
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R = {z C : z = z}z
z
L/K L K : L
L KL/K Aut(L/K)
Aut(C/R) (i)2 = (i2) = (1) = 1 (i) = 1(x + iy) = x iy x, y R Aut(C/R) = {, }
|Aut(Q(i)/Q)| = 2
Q(
3)/Q (x + y
3) = x + y(
3)x, y Q (3)2 = (3) = 3 (3) = 3 |Aut(Q(3)/Q)| 2
3, 3 Q : Q(
3) Q(3) = Q(3) 3 3
K L = K(X)
a bc d
= g
GL2(K) K L L
f(X) f
aX+ b
cX+ d
.
LL K
GL2(K) Aut(L/K)
a 00 a
: a K
Aut(L/K) = GL2(K)/ a 00 a : a K = P GL2(K).L = Q( 3
2) K = Q : L L L Q
( 3
2)3 = (2) = 2 L R 2L ( 3
2) = 3
2 Aut(L/K) = {} L/K
L S L S
LS = {x L : (x) = x S}L L/K
K = LAut(L/K) x
L\
K
Aut(L/K) (x)= x
Gal(L/K) Aut(L/K) L/K
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K LAut(L/K)
CAut(C/R) = R zR z = z
Q(i)Aut(Q(i)/Q) = Q
Q(
2)Aut(Q(
2)/Q) = Q Q(
2) (x +y
2) = x y2 x + y2 y = 0Q
Aut(Q( 3
2)/Q) = {} Q( 32)Aut(Q( 3
2)/Q) = Q( 3
2)
K char K = 2 bK = F2() L = K(x) x
2 = b Lf(X) = X2
b [L : K] = 2 : L
L K
(x)2 = (x2) = b (x) = x = x char K = 2 X2b = (Xx)2Aut(L/K) = {}
L/K Aut(L/K)|Aut(L/K)| [L : K] L/K
1, . . . , nL L
y1, . . . , yn L x L
y11(x) +
+ ynn(x) = 0
y1 = = yn = 0
L G1, . . . , n : G L
L
G = L LL
L
n 1, . . . , n : G Ln > 0
y1, . . . , yn L g G
y11(g) + + ynn(g) = 0
y1, . . . , yn = 0 n > 1 n = 1 y11(g) = 0 1(g) =0 L
h G g gh 1, . . . , n
y11(h)1(g) + + ynn(h)n(g) = 0
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1(h)
y22(g) + + ynn(g) = 0
yi = yi(i(h) 1(h)) 2 i n g Gn y2 = = yn = 0 y1, . . . , yn = 0 i(h) = 1(h)
h G 1, . . . , nL/K |Aut(L/K)| [L : K]
x1, . . . , xn L L/K n = [L : K] 1, . . . , mK L m > n m n
(i(xj)) m > ny1, . . . , ym L j = 1, . . . , n
y11(xj ) + + ymm(xj ) = 0x L x = ni=1 aixi ai K
mi=1
yii(x) =m
i=1
nj=1
yii(ajxj)
=n
j=1
aj
mi=1
yii(xj )
= 0,
L GL [L : LG] = |G| L/LG G
[L : LG] <
K = LG m = |G| L/K [L : K] mm = |G| Aut(L/K) [L : K] m G Aut(L/K)
m = [L : K] G = Aut(L/K)L/K G K LAut(L/K) LG K
x L {x1, . . . , xd} = {g(x) : g G} 1 d m x1 = x xi = xji = j x G f(X) = di=1(X xi)
G f K[X] xK [K(x) : K] m
L/K
K K K/KK = K(x) x L
[K : K] mK [K : K] y L K(y)
K y K [K(y) : K]
m
[K : K] K(y) = K y K K = L [L : K] m
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L/K G
{x1, . . . , xd
}=
{(x) :
G
}G x
L f(X) =
di=1(X xi) f K[X] x Kf x d K
L
L/K K = LG
fG G 1 f
G K[X] K = LG
L/K [L : K] = |Aut(L/K)|
G = Aut(L/K) |G| = [L : LG
]|G| = [L : K] LG = KL/K
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L/K x Lx K L[X]
L x L f K[X] Lf
L f K[X] LL
x L f K[X] deg f = n L/K f L/K f L[X]
f n L
L/K
L/KL/KL K
== L = K(x1, . . . , xn) fi K[X]
xi L/K fi i = 1, . . . , nf L/K f L x1, . . . , xn
f L/K f= L
|Aut(L/K)| = [L : K]M = L =
L/KM/L L/K
M/KM L K
M/K L LM
M M K
L
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L = K(x) fx K M f L
M f K f(x) = 0 M/K
K L M1 M M1/K M1/K x M1 fM1 M = M1M L K f M
L M M
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M/K G = Gal(M/K)K
L
M M f K M
f L M/L Gal(M/L) = { Aut(M) :|L = L} G
: L Gal(M/L) G : H MH
K
L
M
oo //
H
G
L L L H L LH H
K L L M = [L : L] = (Gal(M/L) : Gal(M/L))K G M {1} G
L H G (L) H 1L K Gal(M/L) G Gal(L/K) =G/ Gal(M/L)
M/L MGal(M/L) = L (L) = LGal(M/MH) = H (H) = H
|Gal(M/L)| = [M : L] |Gal(M/L)| = [M : L]
[L : L] =[M : L]
[M : L]=
|Gal(M/L)||Gal(M/L)| = (Gal(M/L) : Gal(M/L
))
x M, G ( 1)((x)) = ((x)) x 1 (x) x MH (x) MH1
H = Gal(M/L) G G H 1 = H (L) = L L
: G = Gal(M/K) Aut(L/K), |L
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{ G : |L = } = Gal(M/L)
[L : K] =|G|
|Gal(M/L)
|= |Im()| |Aut(L/K)| [L : K]
|Aut(L/K) = [L : K]| L/K |Im()| = |Aut(L/K)| Gal(L/K) = G/ Gal(M/L)
L/K H = Gal(L/K) x L fK G 0 = (f(x)) = f((x)) L/K (x) L
(L) = L G H 1 = H GK = Q M X3 2 Q M = Q( 32, 32, 2 32) =
Q( 3
2, ) = e2i/3 xj = j 3
2 2++1 = 0 X322j=0(Xxj)Q
Lj = Q(xj) L0 = Q(3
2) M = L0() [Lj : Q] = 3 [M : L0] = 2 L0[M : Q] = 6 G = Gal(M/Q) 6 G
{x0, x1, x2} G
(j = 0, 1, 2 (xj) = xj ) = = .
G S3 G = S3S3 = Sym{0, 1, 2} S3 {1} {1, (ab)} = (ab) {1, (012), (021)} = A3
13
2
""iii
iiii M
3
}}|||||||
2
##qqq
qqqq
A3
2
aaa
aaa
(a b)3
||zzzzzzz
oo //
MA3
2
fff
ffff M
(a b)
3{{xxxxxxxx
S3 Q
(a b) MA3 2 QQ() M(a b) = Q(xj) xj (a b)
x0 x1, x2 M(1 2) = Q(x0)
C
f R[X] f Rf C[X]
K/R [K : R] K = R x K1
K/C 2
G |G| = pnm gcd(p, m) = 1 G Ppn
H pn = 1 H p
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|H| |H| = p|H| p2 H Z(H) = {x H : xy = yx y
H} = {1} x Z(H) p H = H/x xH K
H p
H p
xK/C K = C L/K
L/R K/R G = Gal(L/R)C R L R [L : R] P G 2
[LP : R] = (G : P) LP = RG = P 2n
H = Gal(L/C) 2 2 H1 H2 [LH1 : C] = (H : H1) = 2
H ={
1}
L = C K = C
f K[X] n L/Kf x1, . . . , xn L f L = K(x1, . . . , xn)
K G = Gal(L/K) G f((xi)) = (f(xi)) = 0 f (xi) = xi i (x) = x x L
= G Sn Gf K Gal(f /K)
G Sn |G| = [L : K] n!H Sn i, j [n] g H g(i) = jH
f K G
y L {y1, . . . , yd} yG y K
di=1(X yi)
x1 (X xj) xj(x1) G f
x1 xj(x1) G
deg f = 2 G = {1} f G = S2 fdeg f = 3 f(X) = (X a)g(X) a K L
g K 2 |G| = 1 |G| = 2f G = Gal(f /K) S3 G = S3
G = A3
(X1, . . . , X n) =
1i
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Disc(X1, . . . , X n) = 2 = (1)n(n1)/2
1i,jn
i=j
(Xi Xj )
(X(1), . . . , X (n) = sgn()(X1, . . . , X n)
f =n
i=1(X xi) L
f = (x1, . . . , xn) L \ {0}Disc(f) = 2f
Disc(f) KK(f) = L
GAn G = Gal(f /K) = Gal(L/K)
f
K Disc(f) K G
An
G (f) = ((x1), . . . , (xn)) = sgn()f(Disc(f)) = (f)
2 = Disc(f)Disc(f) K (f) = f G An
f = 0Disc(f)
Disc(f) =
1i
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p pF p
|F| = pn n = [F : Fp] = dimFp F F pn 1 p : F F, x xp f
F
n 1 Fpn pnfn(X) = X
pn X Fppn
|F| = pn x F xpn1 = 1 x F xpn = xfn(x) = 0 fn p
n F F
F fn
F fn Fp F
p p Aut(F/Fp) F np F = {x F : xpn
= x}fn F
F = F
pn
F F Fp F F F FF = F
fn(X) = pnXpn1 1 = 1 gcd(fn, fn) = 1 fn
p
n
Fpn Fpp
Fpn pm m | n
m n Gal(Fpn/Fpm) = mp
fn pn Fp
nfn = 1 Fpn
Fpp Gal(Fpn/Fp) = G n = [Fpn : Fp] = |G|
p G n mp = m 1 m < n
xFpn x
pm = x
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Fpm Fpn r = [Fpn : Fpm] Fpn = (Fpm)rpn = (pm)r m | n r = mn
m | n H = mp Gal(Fpn/Fp) |H| = mn (Fpn)Hm = (Gal(Fpn/Fp) : H) Fp p
m
Fp
f Fq[X] q = pn deg(f) = d Lf Fq L = Fqm m 1 G = Gal(f /Fq) = Gal(Fqm/Fq)
Sd x1, . . . , xd L f G q = npSd G
d
f = f1f2
fr f
Fq[X] di = deg(fi) qd1, . . . , dr
Si fi i = 1, . . . , r fi qSi q f
Si q (d1, . . . , dr)
f = Xd1 Fp d 1 p d f = dXd1 gcd(f, f) = 1f
Gal(f /Fp) Ad Sd
p
f Z[X] f = gh g, h Q[X]g, h Z[X] Z[X] f
f Q f Z
p p f = gh Fp[X] ff
f(X) = X4 + 5X2 2X 3
X4 + X2 + 1 (X2 + X+ 1)2 (mod 2)X4 + 2X2 + X X(X3 + 2X+ 1) (mod 3)
f f = gh deg g = 1 deg g = 22 3
f
Z[X] p f f
f (mod p)
Sn n = deg f Gal(f /Q) Gal(f /Fp)
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Gal(f /Q) Gal(f /Fp) L = Q(x1, . . . , xn)f(X) =
ni=1(X xi) N = [L : Q] G = Gal(L/Q) = Gal(f /Q)
Sn {x1, . . . , xn} R = Z[x1, . . . , xn]R Z N
L R/pR pN R/pR =ZN/pZN P1, . . . , P m R pR
R/pR k = R/P1pl : R k = R/P1
xi = (xi)
f(X) = f(X) =n
i=1
(X xi) =n
i=1
(X xi)
k = Fp[x1, . . . , xn] R k f FpR/Pj f Fp
|R/Pj | = pl j = 1, . . . , mG R P1, . . . , P m H = StabG(P1) = {
G : (P1) = P1} H R/P1 = k x y mod P1 H(x) (y) mod P1 = (P1) (xi) = xj (xi) = xj xi xi mod Pi
H Gal(f /Fp) Sn {x1, . . . , xn}H = Gal(f /Fp) Gal(f /Fp) G
R1 I1, . . . , I m R i, j i = j
Ii +Ij = R : R R/I1 R/ImI1 Im R/(I1 Im) = R/I1 R/Im
I1 Im m = 2I1 + I2 = R bi Ii
i = 1, 2 b1 + b2 = 1
a1, a2 R x R x ai mod Ii i = 1, 2 bi 0 mod Ii i = 1, 2 b1 = 1 b2 1 mod I2 b2 1 mod I1
x = b2a1 + b1a2 x
Ij = Pj i
= j Pi Pi + Pj
R
Pi Pi + Pj = R
plm = |R/P1 R/Pm|= |R/(P1 Pm)| |R/pR| P1 Pm pR= pN
Nm l
(G : H) G P1(G : H) M
|H| |G|m = Nm l.
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H Gal(f /Fp) = Gal(k/Fp) l |H| = lH = Gal(f /Fp)
P1
Pm = pR G
{P1, . . . , P m}f = g1 gr gi Fp[X] di
Gal(f /Q) (d1, . . . , dr)
K/Fp f gip di p f (d1, . . . , dr)
Gal(f /Q)(d1, . . . , dr) p
f = g1
gr gi di
(d1, . . . , dr) = (1, . . . , 1) pf Fp
f(X) = X4 + 5X2 2X 3
(X2 + X + 1)2 (mod 2)
X(X3 + 2X+ 1) (mod 3)
f Gal(f /F3) 3 X3 X + 1
Gal(f /Q) 3 (1 2 3) = fGal(f /Q) i = 1, 2, 3 (4) = i
1 3 i Gal(f /Q)3 Gal(f /Q) A4 Gal(f /Q) A4 S4
f 2Gal(f /Q) = S4
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K m 1
m(K) = {x K : xm = 1}
m 1 K m(K)m x m(K) m 1 x m
m(K) = x = {xi : 0 i m} m
Xm
1 mXm
1
f mK char K = 0 char K = p m char K = p p | mm 1 K
char K = 0 char K = p, p mf
L Xm 1 m(L) Xm 1 Lm m(L) m 1
L = K()
: G = Gal(L/K) (Z/mZ)() = a (mod m) () = a
GL
Xm 1 =
xm(L)(X x) =
m1a=0
(X a)
L = K()
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m 1 L {a : gcd(a, m) = 1} a = ba b (mod m) G () = a a Z gcd(a, m) = 1
= () = a 1 (mod m)() = b
() = (b) = ()b = ab
G (Z/mZ) a (mod m) a gcd(a, m) = 1 G() = a G {a : gcd(a, m) = 1}
K L C m(L) = {e2ia/m} = e2i/m
m
m(X) =
0amgcd(a,m)=1
(X a
)
m(X) K[X] = L[X]G G {a : gcd(a, m) = 1}
m(X) K
m K m dd | m Xm 1 = d|m d(X) m
1(X) = X 1m > 1
m(X) =Xm 1
1d
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a N gcd(a, m) = 1 G() = a a = p p
a =
prii
f Q f(X) | m(X)f Z[X] g Z[X] p g(p) = 0f(X) | g(Xp) p f(X) | g(XP) = g(X)p f
Xm 1 Fp[X] f(X) | g(X) f2 | fg f = gf g | Xm 1 f2 | fg | Xm 1 Xm 1 f = g
f(p) = 0 G () = p
Q
m 1 m L = Q() G = Gal(L/Q) (Z/mZ) a mod m () = a
[L : Q] = (m) = degm mIm() p mod m p p m
R = Z[] P Z[]p k = R/P = Fp() R kXm 1 Fp : p
p mod m Im()
n cos 2n
n n
2 22k
+ 1
= e2i/n cos 2n =12 ( +
1)n
[Q() : Q(cos 2n )] = 2 cos2n
[Q() : Q] 2 [Q() : Q]
2 G = Gal(Q()/Q)G = H0 H1 Hm = {1} (Hs : Hs+1) = 2
Q() = Fm Fm1 F1 F0 = Q,Fr = Q()
Hr [Fr : Fr1] = 2 cos 2n[Q() : Q] = (n) = |(Z/nZ)| 2
n =
peii pi ei 1
(n) = |(Z/nZ)| =
i
|(Z/peii Z)|
=
ip
ei
i pei
1
i
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=
i
pei1i (pi 1).
p = 2 pe1 2 p pe1(p 1)2 e = 1 p = 2m + 1 m m = rs r, s > 1 r2rs + 1 = (2s + 1)(2(r1)s 2(r2)s + 2s + 1) m
2
22k
+ 1 Fk = 22k + 1 F1 = 5 F2 = 17
F3 = 257 F4 = 65537 Fk k 1641 F5 k 5
Fk
p a Z (a, p) = 1a
p
=
+1 a (mod p)
1 a (mod p)
(Z/pZ) p 1 (p 1)/2 1 p 1
a
p
= a(p1)/2 (mod p),
1p
= (1)(p1)/2 =
+1 p 1 (mod 4)1 p 3 (mod 4) .
abp
=
ap
b
p
p = q
p
q
q
p
=
1 p q 3 (mod 4)+1
fq(X) = Xq 1 Fp[X] L = Fp()
Gal(fq/Fp) Aq
Disc(fq) Fp fq = qX
q1 fq{i : 0 i q 1}
Disc(fq) = (1)q(q1)/2q1i=0
fq(i) = (1)(q1)/2
q1i=0
qi(q1)
= (1)(q1)/2qq(q1)/2(q1)q = (1)(q1)/2qq.
qq1 Disc(fq) Fp
(
1)(q1)/2q
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p 1, , . . . , q1
p(1) = 1 p() = p 1 (q 1)/m
m p Fq p (m 1)(q 1)/m(q
1)/m p Fq
K(x) xm KQ( 3
2) X3 2 Q( 32, e2i/3)
K m 1 char K = 0 char K = p (p, m) = 1 Km
L = K(x) xm = a K L/KXm a [L : K] d 1 xd K Gal(L/K)
Xm a = Xm xm = m1i=0 (X ix) L/K f(X) =Xm a f = mXm1 f L/K
Gal(L/K) f((x)) = 0 (x) = ix i () = (x)/x =i m(K) m : Gal(L/K) m(K) =Z/mZ
, Gal(L/K) (x) = ix (x) = jx ((x)) =(jx) = j(x) = i+jx () = ()()
() = 1 (x) = x = Gal(L/K) m(K) Gal(L/K)
n 1 xn K (xn) = xn xn K ()n = 1 Im n(K) Im = d(K) d
xd KXm a K[X] a d
K d | m d = 1
L = K(x) xm = a Xm a [L : K] = mxm/d
K d
|m d
= 1 a d
K K mL/K m L = K(x) xxm = a K
G = Gal(L/K) = {i : 0 i m 1} y L
x = R(y) = y + 1(y) + + (m1)m1(y)
(x) = (y) + 12(y) +
+ m+1m(y)= x
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(xm) = (x)m = xm
a = xm Ky
L x = R(y)
= 0 i(x) = ix
= x 0 < i < m
L = K(x)
i(X i(x)) Gal(L/K)
K m
Q( 3
2)/Q Q3 Q(cos 27 ) = Q(+
1) = e2/7
Q()
2
6 Q( + 1)
3
Q
Gal(Q()/Q) = F7 = Z/6ZX4 + 4 Q[X] 4
Q
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L/K n L K n x L
Tx : L L, Tx(y) = xyK
x
TrL/K(x) = tr(Tx), NL/K(x) = det(Tx).
fx,L/K x Tx
e1, . . . , en L/K (aij)K xej =
ni=1 aijei j
TrL/K(x) =n
i=1
aii
NL/K(x) = det(aij)
fx,L/K = det(IX (aij))
L = K(y), y2 = d K, y K {1, y}x = a + by Tx
a bdb a
xy = ay + by2 = bd + ay TrL/K(x) = 2a NL/K(x) = a2 db2x, y L, a K
TrL/K(x + y) = TrL/K(x) + TrL/K(y) NL/K(xy) = NL/K(x)NL/K(y)NL/K(x) = 0 x = 0
TrL/K(ax) = a TrL/K(x) NL/K(ax) = a[L:K]NL/K(x)
TL/K: L K NL/K: L K
tr(A + B) = tr A + tr B det(AB) = det(A)det(B)Tx+y = Tx + Ty Txy = TxTy
Tx x L
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Tax = aTx
L = K(x) f(X) = Xn + cn1Xn1 + + c1X+ c0x K fx,L/K = f TrL/K(x) =
cn1
NL/K(x) = (1)nc0
{1, x , . . . , xn1} L/K Tx
0 0 . . . 0 0 c01 0 0 0 c10 1 0 0 c2
0 0 1 0 cn20 0 . . . 0 1 cn1
f fx,L/K = f det(Tx) = (1)nc0 tr(Tx) =cn1K p > 0 L = K(x) xp K, x K
[L : k] = p
y L y K NL/K(y) = y[L:K] = yp TrL/K(y) = [L : K]y = 0y inK y = bixi yp = bpi (xp)i K L = K(y) y
Xp yp NL/K(y) = yp TrL/K(y) = 0TrL/K(y) = 0 TrL/K
L/K TrL/K TrL/K: L K KTrL/K = 0 TrL/K(L) = K
L/K n M L/Kn K 1, . . . , n : L M x L
TrL/K(x) =n
i=1
i(x), NL/K(x) =n
i=1
i(x).
L/K L = K(y) y K My
{e1, . . . , en} L/K P = (i(ej))KA = (aij) aij K Tx
Txej = xej =n
r=1
arj er
i(x)i(ej ) =n
r=1
i(er)arj
SP = P A
S = diag{
i(x)}
P AP1 = S P A{
i(x)}A
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M/L/K x M
TrM/K(x) = TrL/K(TrM/L(x)), NM/K(x) = NL/K(NM/L(x))
{u1, . . . , um} M/L {v1, . . . , vn} L/K Tx,M/L(aij) aij L TrM/L(x) =
mi=1 aii (i, j) Aij
Taij,L/K
TrL/K(TrM/L(x)) =m
i=1
TrL/K(aii) =m
i=1
tr(Aii).
{u1v1, . . . , u1vn, u2v1, . . . , u2vn, . . . , umvn} M/KTx,M/K
A11 . . . A1m
Am1 . . . Amm
m
i=1 tr Aii
L/K
L = K(x) x K TrL/KTrL/K L/K
x = 0 x = 0 n = [L : K] x1, . . . , xnK x k
0
TrL/K(xk) = xk1 + + xkn = 0
f(T) =i(1 xiT)
f(T)f(T)
= T1
k=1
pkTk
pk = xk1 + + xkn f x1, . . . , xn
k pk = 0L/K TrL/K = 0 p = char K >
0 x L K g(Xp
)g(X) K[X] xp K
g(X) L K(x) K(xp) K [K(x) : K(xp)] = pTrK(x)/K(xp) = 0
TrL/K(y) = TrK(xp)/K(TrK(x)/K(xp)(TrL/K(x)(y))) = 0.
L/K TrL/K
M/L/K M/KM/L L/K
L = K(x1, . . . , xn) K L/K
x1, . . . , xn K
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TrM/K = TrL/K = TrM/L
K = K(x1, . . . , xn1) K L = K(xn)K xn K
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f(X) Q[X]f(X) n
L/K K = K0 K1 Kr = L Ki = Ki1(xi) xmi Ki1 m 1i = 1, . . . , r m m
i m = 2f K[X] f
L/K f
M/L L/K M/K
2
deg f = 2 f(X) = X2 + aX+ b a, b Kf f a2 4b = Disc(f)Disc(f) = 0 f f K(a2 4b)
char K = 2 Disc(f) = 0 ff f(X) = (X + )2 K
char K = 2 char K = 2 fchar K = 2 L/K 2 L = K()
K K()
3
char K = 2, 3 f(X) = X3 + aX2 + bX + c X X 13 aa = 0 f(X) = X3 + bX+ c
K K K() =1 = 3 2 + + 1 = 0 char K = 2 K()/KK() = K(
3)
K Lf K
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GG {1} = Ns < Ns1 < < N1 < N0 = G Ni/Ni+1
0 i < s
GGN G N, G/N G
K < G Hi G{K Hi} K Hi+1 K Hi
K Hi Hi/Hi+1 (K Hi)/(K Hi+1) < Hi/Hi+1
N G
G = G/N > (HiN)/N = Hi = Hi/(N Hi),
Hi/Hi+1 = (HiN)/(Hi+1N) = Hi/(Hi Hi+1N)Hi/Hi+1
N,G/N
{1} = Hr Hr1 H0 = N,{1} = Ks Ks1 K0 = G = G/N
Ki = Ki/N Ki < G N Ki+1 Ki K0 = GKs = N Ki/Ki+1 = Ki/Ki+1
{1} = Hr Hr1 H0 = N = Ks Ks1 K0 = GG
S3 {1} A3 S3S4 {1} (12)(34) V4 = (12)(34), (13)(24), (14)(23) A4 S4A4/V4 = Z/3ZSn An n 5 A5A5 Sn An n 5
K char K = 0 f
K[X] f
K Gal(f /K)
deg f 5 Gal(f /K) = A5 f
L/K M/K L/KM/K L = K(x) M
x K L
L = Kr Kr1 K0 = K Ki = Ki1(xi) xmi Ki1i = 1, . . . , r G = Gal(M/K) M0 = K 1
i
rMi = Mi1({(xi) : G})
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Mi Ki Mi/K Mi/Mi1 (xi)m = (xmi ) (Ki1) Mi1
Mi
1/K char K = 0 Mi/Mi
1
M/K
Gal(f /K) Lf K G = Gal(L/K) m = |G|
K m {1} =Hr Hr1 H0 = G Hi/Hi+1 m Ki = LHi
L = Kr Kr1 K0 = KKi+1/Ki Hi/Hi+1 Ki+1 = Ki(xi)
xmi+1
Ki m
K L/K
K = K(m) m m K
Xm 1 m = |Gal(L/K)| Gal(f /K)Gal(f /K) L = L(m) K
K/K L/Kf
f KL/K
f Gal(f /K) Gal(L/K)
Gal(L/K)
L = Kr K1 K0 = K,Ki = Ki1(xi) xmi Ki1
m K Ki/Ki1Hi = Gal(L/Ki) {Hi} Gal(L/K)
L = L(m)
L
nnnnnnnnnn
Ki = Ki(m)
K0 = K(m) = K
K
m m Gal(L/K)
Ki/Ki1 Gal(K
/K)Gal(L/K) Gal(L/K)
4
f
K[X] char K= 2, 3 f
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L f K G = Gal(L/K) = Gal(f /K) S4f(X) =
4i=1(X xi) xi L
S4 a = {12}{(34} b = {13}{24} c = {14}{23}{1, 2, 3, 4} 2 S4 (12) (b, c) : S4 S3 2
V4 = {, (12)(34), (13)(24), (14)(23)} = Z/2ZZ/2ZS4/V4 = S3
L
LGV4 = F
K
L/F Gal(L/F) = V
G
V
F/K Gal(F/K) = G/(G V) = (G) S3F/K
f(X) = X4 + aX2 + bX + c X3
X X+ a x1 + x2 + x3 + x4 = 0 yij = xi + xjy12 = x1 + x2 = (x3 + x4) = y34y23 = y14y13 = y24
G yij {y212, y213, y223} g(T) = (Ty212)(Ty213)(T y223) g LG[T] = K[T]
G G V {12}{34} (yij) = yij (y2ij ) = y2ij y2ij F
y212, y223, y
213 G y2ij
G V F = K(y212, y223, y213)
y212 y213 = (x1 + x2)(x3 + x4) + (x1 + x3)(x2 + x4)= x1x2 + x3x4 x1x3 x2x4= (x1
x4)(x2
x3)
= 0
g(T) = T3 + 2aT2 + (a2 4c)T b2 y12y13y23 = bF = K(y212, y
213) x1 =
12 (y12 + y13 y23) L = K(y12, y13) y212, y213 F
f g
X3
z12 = x1x2 + x3x4
z13 = x1x3 + x2x4
z23 = x2x3 + x1x4
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G
z12 z13 = x1x2 x1x3 + x3x4 x2x4
= (x1 x4)(x2 x3) = 0F = K(z12, z13, z23) {zij}
K h g, h
h X3
zij xi
L
LGV4 = F
KF g h G V4
F = K K
3 | |Gal(f /K)| 3 | |G| G S44 | |G| 12 | |G| G A4 S4
K [F : K] = 2 |G| = 2|G V| |G| | 8 GD4 8 2 S4
f(X) Z[X] Gal(f /Q)An Sn n = deg f
Disc(f) f mod p p An
Sn
deg f = l p f mod pGal(f /Q) l Gal(f /Q) = Sl
f(X) =n
i=1(X xi) Q[X] G Sn H Sn P(X1, . . . , X n) Q[X1, . . . , X n]
H
H = An P = n = 4 H = D4 = (1234), (12)(34)P = X1X3 + X2X4
g(Y) =
Sn/H(Y P(x1, . . . , xn))
=
Sn(Y P(x1, . . . , xn))1
|H|
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H G
P Q[Y] Y2 Disc(f) hP Q G H
H Sn P g xi C g
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