tripos ii galois theory

7/31/2019 Tripos II Galois Theory

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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


8/65

: 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


10/65

= (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


11/65

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)


31/65

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


32/65

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


33/65

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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35/65

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


36/65

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


37/65

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


38/65

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


39/65

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


40/65

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


41/65

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


42/65

{ 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


43/65

|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


44/65

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


45/65

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


46/65

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)


47/65

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.


48/65

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


49/65

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()


50/65

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


51/65

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


52/65

=

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


53/65

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


54/65

(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


55/65

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


56/65

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


57/65

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

tripos ii galois theory

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