5/26/2018 Galois

1/74

Galois

Galois

5/26/2018 Galois

2/74

5/26/2018 Galois

3/74

x3 2 xn 1

Galois

Galois

x3 2 Galois Galois Galois Galois

5/26/2018 Galois

4/74

5/26/2018 Galois

5/74

f(x) C[x] deg f(x) > 0 a C f(a) = 0 C

f(x)C[x] f(a) = 0 q(x) C[x] f(x) = (x a)q(x) f(x) f(x) C[x] n= deg f(x)> 0 f(x) n C C C[x] C[x] R[x] C[x]

C R x2 + 1 R[x] R C[x]

5/26/2018 Galois

6/74

Galois

Gauss Argand Gauss

Galois

dal Ferro, Cardano, Tartaglia, Ferrari

x3 +mx n= 0

3

n

2+

(

n

2)2 + (

m

3)3 3

n

2+

(

n

2)2 + (

m

3)3 .

5/26/2018 Galois

7/74

Galois

Ruffini Lagrange

Ruffini

Abel

Abel-Ruffini

f(x) =x5 2 C[x] f(x) b= 5

2 f(x) = (x b)q(x) deg q(x) = 4 q(x)

f(x)

f(x) f(x) Galois Galois f(x)

Galois

Galois

k f(x) k[x] E k f(x) E k Galois f(x)

5/26/2018 Galois

8/74

Galois

k

(0, 0) (1, 0) (2, 0)

(1, 0) (

1, 0)

(n, 0) : n Z

a, b b= 0 ab a b a/b a

{(a, b) : a2, b2 Q}

5/26/2018 Galois

9/74

Galois

k 0= f(x) k[x] f(x) k[x] f(x) = q1(x)q2(x) q1(x), q2(x)k[x] q1(x) q2(x)

x3 2

f(x) = x3 2 f(x) Q[x] R[x] C[x] f(x) C b=

3

2 f(x) m= 0 n= 2 x bR[x] x3 2

f(x) =x3 2 = (x 3

2)(x2 + 3

2 x + 3

4)

f(x) = (x b)(x2 +b x+b2

p(x) =x2 + b x + b2 x3 2

3

2(12i

3

2 ).

5/26/2018 Galois

10/74

Galois

=12

+i

3

2 .

0 =3 =e2i = 1 , = e2i/3 = cos(2

3) +i sin(

2

3) , 2 =1

2i3

2 .

x3 2 C b b 2b

f(x) Q Q[x] f(x) f(x) Q[x]

b q(x) R f(x) =

(x b)q(x) R[x]

f(x)

R[x]

q(x) R q(x) R[x] R[x] f(x) f(x) = (x b)q(x)

f(x) C[x] f(x) = (x b)(x b)(x 2b)

k f(x) k[x]

bk

f(b) = 0

f(x) = (x b)q(x) q(x) k[x]

deg f(x) = 1 f(x)

deg f(x) = 2, 3 f(x) f(x) k

kF F f(x) F[x] f(x) k[x]

xn

1

f(x) =x3 1 Q[x] f(1) = 0

x3 1 = (x 1)(x2 +x+ 1) .

5/26/2018 Galois

11/74

Galois

3(x) =x2 +x+ 1

3(x) 2 = e2i/3 3 (3)

3 =3 = 1 x3

1 3

23

33 = 1

(x 3)(x 23) =x2 +x+ 1

23+3+ 1 = 0

3 23 = 1

3(x) R[x] Q[x] C[x] n n

n= e2i/n = cos(

2

n) +i sin(

2

n)

n n n(x)

n(x) =xn1 +xn2 + +x + 1 .

xn 1 = (x 1)n(x)

n(x) C n 2n

n1n

n

(kn)n1 + (kn)n2 + (kn) + 1 = 0 k= 1, . . . , n 1

n 2n n1n = 1

n(x)

Q[x]

n

n

2/n n= 5

5/26/2018 Galois

12/74

Galois

n = 4 4 = ei/2 = i 4(x) = x

3 +x2 +x+ 1 = (x+ 1)(x2 + 1) 4(x) Q[x]

n(x)

1 n(x) Q[x] n(x) R[x]

k k[x] I k[x]

f1(x), f2(x)

I f1(x)

f2(x)

I

f(x)I g(x) k[x] g(x) f(x)I f(x)k[x] (f(x)) :={q(x)f(x) : q(x)k[x]} k[x] f(x) I= (f(x)) f(x) g(x) (f(x)) = (g(x)) f(x) =cg(x) c k C[x] (x2 + 1) = (2x2 + 2)

k[x] k k k[x]

{xi : i N} dimkk[x] = k[x]

5/26/2018 Galois

13/74

Galois

k[x] I k[x] f(x) k[x] I = (f(x)) I= 0 f(x) I

I = (f(x)) k[x] f(x) f(x) k[x] k[x]/I

k[x] I k[x]/I

k[x] (f(x)) f(x)

f(x) k k[x]/(f(x)) : kk[x]/(f(x)) (c) =c + (f(x))

k[x] f(x), g(x)R f(x)=0 q(x) r(x)R g(x) =f(x)q(x) +r(x) r(x) = 0 deg r(x)< deg q(x)

k[x] 0=f(x) R f(x) f(x)

q1(x), . . . , q s(x) k[x] n1, . . . , ns N qi(x)=qj(x) i=j ni> 0 i= 1, . . . , s

f(x) =q1(x)n1 qs(x)ns . f(x) = g1(x)

m1 gt(x)mt f(x) i= j gi(x)=gj(x) 0< mi N i= 1, . . . , t t= s {q1, . . . , q s}={g1, . . . , q s}

f(x), g(x)k[x] f(x) g(x) k[x] (f(x), g(x)) f(x) g(x) (f(x), g(x)) =

{q1(x)f(x)+

q2(x)g(x) : q1(x), q2(x) k[x]} (f(x), g(x)) f(x) g(x) (f(x), g(x)) = 1 (f(x), g(x)) = k[x]

5/26/2018 Galois

14/74

Galois

I = (x2 + 1) R[x] R R[x]/I 1 :R R[x]/I 1(c) = c+ I R[x]/I f(x) + I f(x) = (x2 + 1)q(x) +r(x) r(x)

Q[x] deg(r(x))< 1

f(x) + I = r(x) + I R[x]/I a+ bx + I a, b R 2: R[x]/I C 2(a + bx+ I) = a + bi 3 = 2 1 : R C 3(c) =2 1(c) =2(c+I) =c R C

I = (x2 3) E = Q[x]/I x2 3Q[x] E f(x) + I f(x) Q[x] f(x) = (x2 3)q(x) + r(x) r(x) Q[x] deg(r(x)) < 2 f(x) + I=r(x) + I x2 + I= 3 + I E (x + I)1 = 13x + I (x+ 2 +I)1 =x+ 2 +I Q E cc + I c +I c

k k[x]

3 Q[x] Gauss

f(x) = anxn + +a0 Z[x]

a0, . . . , an deg f(x)> 0 f(x) Z[x] f(x) Q[x]

f(x) =anxn

+ + a0 Z[x] deg f(x) =n

rs Q (r, s) = 1 f(x) r| a0 s| an

f(x) f(a)= 0 a a0 f(x) Q

5/26/2018 Galois

15/74

Galois

x3 3x 1 Q[x]

x2 p x3 p Q[x] p

Eisenstein

f(x) = anxn + +a0Z[x] pZ

p ai i= 0, . . . , n 1 p an p2 a0 f(x) Q[x]

f(x) = 2/9x5 + 5/3x4 +x3 +1/3 Q[x] f(x) Q[x] 9f(x) = 2x5 + 15x4 + 9x3 + 3 Q[x] Eisenstein 9f(x) p= 3

f(x)

k f(x) =k[x] f(x) g(x) = f(ax +b) a, b k a= 0 Eisenstein

f(x) =x16 +x15 + +x+ 1Q[x] Eisenstein Eisenstein

f(x+ 1) p = 17 Z[x] f(x) Q[x]

pZ p(x) =xp1 +xp2 + +x+ 1Q[x] Eisenstein p(x+ 1) p p(x)(x 1) =xp 1

p(x+1) =(x+ 1)p 1

x =

xp +p1

xp1 +

p2

xp2 + + pp1x+ 1 1

x

=xp1 + p1xp2 + p

2xp3 + +p

Eisenstein q(x+1) p p(x + 1) p(x)

5/26/2018 Galois

16/74

Galois

4(x) =x3 +x2 +x+1 Q[x] C 1, i x + 1 4(x)

f(x)

Z[x]

Zp[x] p Z : ZZp a aa mod p

: Z[x] Zp[x], a0+a1x+ +anxn a0+a1x+ +anxn .

f(x)Z[x] p deg f(x) = deg (f(x)) (f(x)) Zp[x] f(x) Z[x]

Zp (f(x))

x2+x+1 x3 +x+1 Z2[x] Z2 x2 + x + 1 x3 + x + 1 Z[x] Q[x]

x2 +1 Z2[x] x2 +1 = (x+1)2

Z2[x] Z[x]

4 Z2[x] x x+1 x2+x+1 x3 +x+ 1 x4 +x3 + 1 x4 +x+ 1 x4 +x3 +x2 +x+ 1

x6+x3+1 Z2[x] 7x4+5x3+3 x6 + 11x31 Q[x] x4 + x3 + 1 x6 +x3 + 1 Z2[x]

f(x) =x

4

10x + 1 Z[x]

Zp[x] p

5/26/2018 Galois

17/74

Galois

k q(x) f(x) k[x] q(x)| f(x) q(x) k[x] f(x) =q(x)q(x)

f(x) k[x] a k f(x) (x a)| f(x) k[x] f(x) = (x a)q(x) +r(x) deg r(x)

5/26/2018 Galois

18/74

Galois

k p(x) k[x] p(x) k[y]/(p(y))

I = (p(y))

E = k[y]/I

k

E

k E c c+ (p(y)) y+I E p(x) p(x) =a0+a1x + +anxn k[x]

p(y+I) =a0(1 +I) +a1(y+I) + +an(y+I)n

= (a0+I) + (a1y+I) + + (anyn +I) = p(y) +I=I .

F k L/ k k F Kronecker

Kronecker f(x) k[x] k L/k f(x) L[x]

f(x) deg f(x) = 1 L = k deg f(x) > 1 f(x) = g(x)p(x) p(x), g(x)k[x] p(x) p(x) f(x) L g(x) L deg g(x) = deg f(x) 1< deg f(x)

degp(x)> 1 M/ k p(x) aM p(x) = (xa)h(x)M[x] f(x) = (x a)h(x)g(x) M[x] deg h(x)g(x) < deg f(x) L M h(x)g(x) f(x) L k

Kronecker deg f(x) =n L f(x)

f(x) =c(x a1)s1 (x at)st

c k ai L ai= aJ 1 i t f(x) n n = s1+ +st

5/26/2018 Galois

19/74

Galois

s1, . . . , st a1, . . . , at f(x) L f(x)

f(x) =c0+ c1x + + cnxn k[x] k L f(x) f(x) f(x) f(x) = c1+ 2c2x+ +ncnxn1 f(x) L (f(x), f(x))= 1 (f(x), f(x)) k L

f(x) =c(x a1)s1 (x at)st L[x] . si > 1 i {1, . . . , t} xai f(x) f(x) (f(x), f(x))

= 1

g(x)k[x] k[x] g(x) k 0 g(x)= 0 deg g(x) = deg g(x) 1 g(x) g(x) (g(x), g(x)) =1 g(x) g(x) k[x] g(x) f(x) k[x] f(x) f(x)

k f(x)k[x] k f(x)

k p p f(x) f(x) f(x) f(x) f(x) f(x) f(x) = 0 f(x)

f(x) =x2 + 1

Z2[x] f(x) = 0 f(x) = (x+ 1)2

f(x) = xp

n

+xZp[x] p f(x) Zp[x] p f(x) =pnxp

n1 +1 = 1 (f(x), f(x)) = 1

5/26/2018 Galois

20/74

Galois

f(x) = xp x+a Zp[x] p f(x) =1 f(x)

k = Z2[t] ={a(t)/b(t) : a(t), b(t) Z2[t], b(t)= 0} f(x) = x2

t

k[x]

k f(x) = 0 f(x)

k f(x) k[x] Kronecker L/ k f(x) k F L f(x) F[x] L/ k f(x) k

C x2 2 Q x2

2 = (x

2)(x+

2)

R[x] R x2 + 1 Q

Q[x]

f1(x) =x9 + 4x+ 6

f2(x) =x+ 1 f3(x) =x4 + 4 f4(x) = 8x3 6x 1 f5(x) =x4 2x2 + 9 f6(x) =x4 + 1 f7(x) =x7 + 7x+ 14 f8(x) =x(p1)p +x(p2)p + +x2p +xp + 1 p f9(x) = 4/3x5 + 6/5x2 + 2 f10(x) =x5 10x + 2 f11(x) =x5 10x + 1

x4 + 4Z3[x]

5/26/2018 Galois

21/74

Galois

x4 + 4Z13[x] x2 + 3Z7[x]

x2 5 Q(2) Q

Z3

5/26/2018 Galois

22/74

Galois

5/26/2018 Galois

23/74

F/k

F

k

k F F (c, f(x))cf(x) F

F/ k a F a k f(x)

k[x] f(x)

= 0

f(a) = 0 a k a k

f(x) k[x] f(x)F[x]

aF a F f(x) =x aF[x]

a =

3

R Q a

f(x) =x2 3 Q[x]

Q[y]/(y2 3) Q a = y+ (y2 3) x2 3 a Q

5/26/2018 Galois

24/74

Galois

f(x) k[x] I= (f(x)) F = k[x]/I x+ I F k

k

F

E a

E

k a F

i C R Q

z= a + bi C a, b R za= bi z22az+ a2 =b2 z x22ax+(a2+b2) R[x] C R

a=

2 b=

3R Q

x2 2 x2 3 Q[x] ab= 6 Q x2

6 Q

[x]

a + b=

2 +

3 Q

c = a+b =

2 +

3 c2 = 5 + 2

6 c2 5 = 26 (c2 5)2 = 24 c4 10c+ 1 = 0 c f(x) = x4 10x+ 1 f(x) Q[x]

R Q

Lindemann e Q Hermite ei =1

E= k(x) k[x] xE k

F/ k a F a k k a

F/ k a

F k[a] =

{f(a) :f(x) k[x]} k(a) ={f(a)/g(a) :g(a)= 0, f(x), g(x) k[x]} k[a] k a k(a) k a k k[a] k(a) F k(a)

5/26/2018 Galois

25/74

Galois

k[a] k(a) k k[a] E kE aE k(a) E kE a

E

i2l =1 i2l+1 =i lN f(x)R[x] f(i) =a + bi a, b R R[i] ={a + bi: a, bR}= C R[i] = R(i) = C R R[i]

{1, i} dimRR[i] = 2

32l

= 3l

32l+1

= 3l

3 l N

Q[

3] =

{a +b

3 : a, b

Q

}.

a, bQ 0=a2 3b2 Q c= a2 3b2

1

a +b

3=

1

a2 3b2 a b

3 =a

cb

c

3 Q[

3] .

Q(

3) Q[3] Q[3] = Q(3) Q Q[3] {1, 3} dimQQ[

3] = 2

Q[ 3

2] m

N m =

3l+k k, l N l2

( 3

2)m = ( 3

2)3l+k = 2l( 3

2)k

f(x) =

cixi Q[x]

f( 3

2) = a0+a13

2 +a23

22

: ai Q.

Q[ 3

2] ={a0+ a1 3

2 + a23

4 : aiQ} {1, 3

2, 3

4} Q[ 3

2] Q dimQQ[

3

2] 3 dimQQ[

3

2] = 3

E1 = Q[

2] E2 = Q[

2 +

3] E1 E2

2 +

3 = 12 +

3

E2 ,

5/26/2018 Galois

26/74

Galois

2 =

(

2 +

3) (2 + 3)2

E2 . E1 E2 E2 = E1

2 +

3 /

E1

3 = (

2 +3)(2) E1 Q E1 {1, 2}

3 =a+b

2, a, b Q

3 =a2+2b2+2ab

2

2 / Q

p = e2i/p kN (k, p) = 1 Q[] =Q[k] Q[k] Q[] k Q[] r, t Z rp + tk= 1 = rp+tk =rp tk =k

t Q[k] F/k a

F

: k[x] k[a] , (h(x)) =h(a)

(c) = c c k (x) = a ker ={f(x) k[x] : f(a) = 0} k[x]/ ker = Im = k[a] k[a] ker

F/k a F a k k[a] = k(a) a k dimkk[a] =

a k f(x)k[x] f(a) = 0 ker = 0 ker k[a]=k[x]/ ker k[a] k(a) k(a) k a k[a] =k(a) a k ker = 0 k[a]=k[x] k[a] dimkk[a] =

k k[a] a

F/k

F

k [F : k] F k

5/26/2018 Galois

27/74

Galois

[C : R] = 2 [Q[3] : Q] = 2 [k(x) : k] = [Q[] : k] =

Q[]R Q Q[] Q R Q dimQQ[] Q R dimQQ[] = [R : Q] =

k F E [F : k] = [E :k] =

F/k a F k

k[a] = k(a) I={f(x) : f(a) = 0} k[x]

I I= (g(x)) g(x) k[x]

h(x) I h(a) = 0 h(x) = q(x)g(x) h(x)= 0 deg h(x)deg g(x)

h(x) k[x] h(a) = 0 h(x) =cg(x) c k[x]

F/k aF k[x] k[x] a a k

irr(k,a)(x)

5/26/2018 Galois

28/74

Galois

aF irr(F,a)(x) =x a

a=

3

R irr(Q,a)(x) =x2

3

p =e2i/p C irr(Q,)(x) = p(x) =xp1 +xp2 + + 1

k[a] k[x]

F/k a F k deg irr(k,a)(x) = n {1, a , . . . , an1} k k[a]

f(x) = irr(k,a)(x) B ={1, a , . . . , an1} g(a)

k[a]

g(x) k[x] g(x) =f(x)p(x) + r(x) p(x), r(x) k[x] t= deg r(x)< n r(x) =ctx

t + +c1x+c0 ci k i= 0, . . . , t g(a) = f(a)p(a) +r(a) = r(a) = cta

t + + c1a+ c01 g(a) k B B d0 1 + + dn1an1 = 0 di k i= 0, . . . , n 1 ai : 0in 1 g(x) = d0+ d1x+ +dn1xn1 g(a) = 0 g(x) (f(x)) g(x)= 0 deg g(x)< deg f(x) g(x) = 0 di= 0 i= 0, . . . , n 1

F/k a F a k [k[a] : k]

5/26/2018 Galois

29/74

Galois

[F :k] =n a F {1, a , . . . , an} n + 1 d01 + + dnan = 0 di k i = 0, . . . , n g(x) =d0+ d1x + dnx

n

k[x] a g(x)

k

F/k a1, . . . , an F k[a1, . . . , an] k a1, . . . , an k(a1, . . . , an) k a1, . . . , an

k[a1, . . . , an] = k[a1, . . . , an1][an] k(a1, . . . , an) = k(a1, . . . , an1)(an)

Q[2, 3] =Q[2 + 3]

3 Q[2 + 3] Q[2] Q[2 + 3]

3 Q[2 + 3] Q[2][3] =

Q[

2,

3] Q[2 + 3]

2 +

3 Q[2, 3] Q[2 + 3] Q[2, 3]

Q[

2,

3] = Q[

2 +

3].

k F E E F k

k F E

[F :k]

5/26/2018 Galois

30/74

Galois

k {b1, . . . , bm} j = 1, . . . , m dij = 0

E k

k F E a E k irr(F,a)(x) irr(k,a)(x) deg irr(F,a)(x)deg irr(k,a)(x)

E = Q[

2 +

3] Q E

Q Q[2] E {1, 2, 3, 6} f(x) = x410x + 1

2 +

3 f(x)

Q[x]

4 = [Q

(

2 +

3) :Q

] = deg irr(Q,

2+

3)

(x)

irr(Q,2+

3)(x) =f(x) Q E

{1, 2 + 3, (2 + 3)2, (2 + 3)3}

b = 3

2 = e2i/3 E = Q[b, ] irr(Q,b)(x) = x3 2 [Q[b] : Q] = 3 {1, b , b2} Q Q[b] irr(Q,)(x) = x

2 +x + 1 irr(Q[b],)(x) irr(Q,)(x) 2 / Q[b] deg irr(Q[b],)2

irr(Q[b],)(x) = irr(Q,)(x) =x2 +x + 1

{1, } Q[b] E [E :Q] = 6 Q E {1, b , b2,,b,b2}

b= 5

2 = e2i/5 E= Q[b, ]

[E: Q] = [E : Q(b)][Q(b) : Q]

deg irr(Q,b)(x) = 5 5 [E : Q]

[E : Q] = [E :Q()][Q() : Q]

deg irr(Q,)(x) = 4 4 [E : Q] [E : Q] [E : Q] 20 [E : Q(b)] = deg irr(Q(b),)(x)

deg irr(Q(b),)(x)deg irr(Q,)(x) = 4 .

5/26/2018 Galois

31/74

Galois

[E : Q] 20 [E : Q] = 20 irr(Q(b),)(x) =x4+x3+x2+x+1 irr(Q(),b)(x) =x5 2

kE F k F E k E [E : k] = p p F k F E E k

a E a / k k k[a] k[a] =E

R[x]

f(x) R[x] aC f(a) = 0 aR (xa)| f(x) f(x) x a= cf(x) cR f(x) a /R [C : R] = 2 R[a] =C f(x) = irr(R,a)(x) deg f(x) =[R[a] : R] = 2

Q p= 0 Zp p p

F/Zp Zp [F : Zp] = n F Zp n Zp a1, . . . , an F F ={c1a1 + + cnan : ai Zp, 1in}

F= Zp Zp Zp .

|F|=|Zp|n , |F|= pn .

5/26/2018 Galois

32/74

Galois

F p F p |F| =pn n1 (F, ) F F = F {0}

pn

1 Lagrange

aF ap

n1 = 1apn =aapn a= 0 , a

f(x) =xpn x Zp[x] .

F |

F|

= pn p F f(x) =xp

n x F f(x) F pn

n > 1 p F p f(x) = xp

n x Zp[x] f(x)

Kronecker L/Zp

f(x) = xpn x M f(x) L

M={aL : apn =a}. M L a, bM p pni 1ipn 1 p = 2

(a b)pn =apn bpn =a ba bM . a, bM bM {0}

(ab1)pn

=apn

(bpn

)1 =ab1 ab1 M . M L M f(x) pn f(x) M pn

p n >1 pn

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33/74

Galois

E l N 8 = 4l E

F |F| = 4 = 22 F F Z2 F ={0, 1, a , b} a+ 1F a+ 1 =b a + 1 =a 1 = 0 a + 1 = 1 a= 0 a+ 1 = 0 a=1 a= 1 (F, +)

0 1 a a + 1 =b0 0 1 a b1 1 0 b a

a a b 0 1b= a + 1 b a 1 0

(F, ) F a b F={1, a , a2 =b} (F, )

1 a a2 =b1 1 a ba a b 1

b= a2 b 1 a

.

a2 =b

a2 =a+ 1a2 a 1 = 0a2 +a+ 1 = 0 , 1 =1 Z2 a x2 + x +1Z2[x] x2 +x+ 1 F = Z2(a) irr(Z2,a)(x) =x

2 + x + 1 x2 + x + 1 aa2 =a + 1

x8 x = x23 x Z2[x] F

F (F, ) F F 1= b F F 0, 1=bF F

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34/74

Galois

Galois

b F = Z2(b) 0, 1= b F [F :Z2] = 3 x8 x Z2[x]

x8

x= x8 +x= x(x+ 1)(x3 +x2 + 1)(x3 +x + 1) .

aF x3 + x2 + 1 irr(Z2,a)(x) =x3 + x2 + 1 Z2(a) =F 1, a , a2 Z2 F

F ={c0+c1a+c2a2 : ai Z2, 0i2}={0, 1, a, 1 +a, a2, 1 +a2, a +a2, 1 +a +a2}.

irr(Z2,a)(x) =x3 +x2 + 1

a3 =a2 + 1

a4 =aa3 =a2 +a + 1

a5 =a2a3 =a + 1

a6 =aa5 =a2 +a

a7 = 1

x3 + x2 + 1 a, a2, a4 x3 + x + 1 a3, a5, a6 a F x3 +x+ 1

Galois

F/k F k

Autk(F) ={: F F, (c) =c, c k} Galois F k Gal(F/k)

G= Gal(F/k)

f, gG f g :FF c k fg(c) = f(g(c)) = f(c) = c fg G f G f1 : FF c k f1(c) = f1(f(c)) =(f1 f)(c) = idE(c) =c f1 G G F F G

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35/74

Galois

a F k G = Gal(F/k) a (a) (a) irr(k,a)(x)

F/k a

F k Autk(F) (a) = b bF k irr(k,a)(x) = irr(k,b)(x)

q(x) = irr(k,a)(x) =

cixi

a q(x) = 0

cia

i = 0 0 =(

ciai) =

(cia

i) =

(ci)(ai) =

ci(a)i =

cibi

q(x) = irr(k,b)(x)

F/k a, bF k irr(k,a)(x) = irr(k,b)(x) : k(a)k(b) |k= idk (a) =b

k(a) =k[a] k(b) =k[b] I k[x] irr(k,a)(x) I = (irr(k,a)(x)) 1 : k[x] k[a] 1(f(x)) = f(a) 1 :k[x]/I k[a] 1(f(x) + I) = f(a) 1(x+I) = a 1(c+I) =c c k 2: k[x]/I k[b] 1(f(x) +I) = f(b) 211 :k[a] k[b]

: k k

: k[x] k[x], aixi (ai)xi

k F kF bF bF k k : k k (irr(k,b)(x)) = irr(k,b)(x) . :k[b]k[b] |k = (b) =b

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Galois

Galois

: k[x] k[x] k[x] I= (irr(k,b)(x)) k[x]

1: k[x]

k[x]/I, f (x)(f(x)) +I

J= (irr(k,b)(x))

2 : k[x]/J k[x]/I, f (x) +J(f(x)) +I .

3 : k[b] k[x]/J, f (b)f(x) +J

4: k[b] k[x]/I, g(b)g(x) +I = 14 23

Gal(F/k) F/ k idF Gal(F/k) F = k(a1, . . . , an) Gal(F/k) b= f(a1, . . . , an)F f(x1, . . . , xn) k[x1, . . . , xn]

(b) =(f(a1, . . . , an)) = f((a1), . . . , (an)).

(a1), . . . , (an)

Gal(Q/Q) = AutQ(Q) ={idQ}

k Gal(k/k) = Autk(k) ={idk}

G= Gal(C/R) G=Z2 C = R(i) f(x) =x2 + 1 R G R

i

f(x)

G R(i) =C G G={0, 1} 0: ii 1 :i i 0=idC 1(a +bi) =a bi

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37/74

Galois

F =Q() =e2i/3 irr(Q,)(x) = x2 +x+ 1 F c+d : c, dQ Gal(F/Q) (c) =c,c Q

() = 2 F = Q(2) : F Q(2) () = 2 Gal(F/Q)= Z2

E=Q(2) ={c0 1 +c1

2 : ciQ} G= Gal(Q/E) irrQ

2 =x2 2 (2) =2

G a E

a= c0 1 +c1

2 .

(a) =(c0 1 +c1

2) =(c0 1) +(c1

2)

=(c0) (1) +(c1)(

2) =c0 1 +c1(

2) .

G (2)

|G| 2 idE(

2) =

2 : E Q(2) (

2) =2 E=Q(2) G

G |G|= 2 G=Z2

E=

Q( 3

2) G= Gal(Q

/E) b= 3

2 irr(Q,b)(x) =x3 2 Q E {1, b , b2} a EG

a= c0+c1b+c2b2 : ci Q

(a) = (c0) +(c1)(b) +(c2)(b2) =c0+c1(b) +c2(b)

2 .

G (b) (b) irr(Q,b)(x) x3

2 E b (b) =b = idE

G={idE}

E = Q(

2,

3) G = Gal(Q/E) G (

2) (

3)

G G (2)

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Galois

Galois

2 (3) 3 idE G

2 2 3 3

G E E1= Q(

2)

E=E1(3) . 1

1 : E1(

3)E1(

3), 1(

3) =

3, 1(c) = c, cE1 .

E1(

3) =E1(

3) 1G

1(

2) =

2, 1(

3) =

3 .

E E2 = Q(

3) 2

2(

2) =

2, 2(

3) =

3 .

3 = 1 2G, 3(

2) =

2, 3(

3) =

3 .

3 E1 = Q(

2)

: E1E1, 1(

2) =2, 1(c) =c, c Q.

(a + b

2) =a b2 : E1(

3)E1(

3)

3

3 3

|G| = 4

Z4 Klein Z2 Z2 G

21:

2

2

2,

3

3 (

3) =

3,

21 = idE 1 G= Z2 Z2

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39/74

Galois

: k k L/ k f(x) L/ k f(x) : L

L

G= Autk(F)

F

G

G

Q(

5,

7 Q(i

11) Z2(x2) Z2(x+ 1) Z2(x2 + 1)

irr(Q,a)(x)

a= 7 + 1/2

a= i

3 1/2

irr(Z3(x),a)(x) a2 =x+ 1

a, b C irr(Q,a)(x) =x22 irr(Q,b)(x) =x24x+2 Q(a)/Q Q(b)/Q

Q(a)/Q irr(Q,a)(x)

x2 5

x4 +x3 +x2 +x + 1

x3 + 2 Galois Gal(Q(a)/Q)

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Galois

C/Q Z5(x)/Z5 R(5)/R

Q( 7

3)/Q

E= Q(

2,

3,

5) [E : Q] = 8

f(x) =x3 +x2 + 2x

E f(x) Z3[x]

E

a E= Z3(a)

a, b E F

[F(a +b) :F] a +b F

Q(

5 +

2) = Q(

5,

2)

5+

2 Q(

2)

5+

2

Q(

5,

2)

5 +

2 Q

Z5

F f(x) =x315 x

Z3 E f(x) 315 Z3 [E: Z3]

L/ k f(x) k[x] deg f(x)>1 [L: k] f(x) L

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Galois

x3 2

f(x) = x3 2 Q[x] f(x) C b,b,2b b = 3

2 = 3 = e

2i/3

E = Q(b,b,2b) f(x) Q

E= Q(b,b) = Q(b, 2b) = Q(b,2b) = Q(b, ) .

G= Gal(E/Q) G (b) () G E= Q()(b)

irr(Q(),b)(x) = irr(Q,b)(x) =x3 2, E cc c Q() b x3 2

b

b

b

2b

.

E= Q(b)()

irr(Q(b),)(x) = irr(Q,)(x) =x2 +x+ 1,

E cc c Q(b)

2 .

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Galois

Galois

b b c c cE E G

b b b 2b b b 2b

2 2 2idE 1 2 3 4 5

,

2 =21 4=13 5 = 31 G

aE 1(a) Q E {1, b , b2,,b,b2}

a= a0+a1b+a2b2 +a3+a4b +a5b

2

1(ai) =ai 1(b) =b 1() = 1(b2) =2b2 1(b) =

2b 1(b

2) =3b2 =b2 x2 + x + 1 2 =

1

1(a) =a0 a4b+ (a2+a5)b2 +a4+ (a1 a4)b a2b2 .

|G| = 6 S3

X={b,b,2b} G{(b), (b), (2b)}= X .

1(b) =b 1(b) =1()1(b) =2b 1(

2b) =3b=

b x32 x32

G E G X X SX X : GSX G ()SX

() : XX, ib(ib) . () G

idE b b 2bb b 2b , 1 b b 2bb 2b b , 2 b b 2b2b b b ,3

b b 2bb 2b b

, 4

b b 2b

b b 2b

5

b b 2b2b b b

.

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Galois

SX , G

( ) =() ()

(i ) (i) () tb

G=SX . G

tb

0 2/3 4/3 idE 1 2

3 4 5

Galois

f(x) k[x] deg f(x) =n E f(x) Gal(E/k) Sn

X ={

b1, . . . , bn}

f(x) E = k(b1, . . . , bn) SX X SX=Sn G= Gal(E/k) G : X X, (bi) = (bi) i = 1, . . . , n SX

(bi) =(bj)bi = bj

: GSX, .

( ) = = =() () , G () =() = , G

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Galois

Galois

Gal(E/k) Sn

f(x) = x4

2 b= 4

2 f(x)

C b bi E= Q(21/4, i) f(x) G= Gal(E/Q) G G

(b) =

b

bib

ib

, (i) =

i

i .

G

: b ib,i i

: b b, i i

G

b b b b b ib ib ib ibi i i i i i i i i

idE .

b ib b ibib b ib b

b ib b ibb ib b ib

.

f(x) b, ib, b, ib 3/2 =2/4 3 b, b

Galois

: E1

F1

: E1[x]F1[x], aixi (ai)xi

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45/74

Galois

: E1F1 f(x) =

aixi

E1[x] E f(x) E1 F

(f(x)) F1 [E : E1]

: EF |E1 = [E :E1]

[E : E1] = 1 E = E1 f(x) E (f(x)) F1 F =F1

[E :E1] n

[E : E1] = n bE\ E1 f(b) = 0 E1(b)=E1

[E :E1] = [E :E1(b)][E1(b) :E1]

[E :E1(b)]< n b

(f(x)) F : E1(b)F1(b) |E1 = : E F |E1(b) = [E :E1(b)] b f(x) b f(x) f(x) deg f(x) deg f(x) = [E1(b) :E1]

[E :E1(b)][E1(b) :E1]

[E :E1]

f(x)

f(x) F[x] E, E f(x) F : E E F f(x) [E : F] F

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Galois

Galois

idF :F F

(f(x)) = f(x) E=E

Gal(E/F)

f(x) F[x] E f(x) F | Gal(E/F)|= [E :F]

E= Q(

2,

3,

5) G= Gal(E/Q) E f(x) = (x2 2)(x2 3)(x2 5) [E :Q] = 8 |G| = 8 G

2 2 3 3 5 5 G G

G= Z2 Z2 Z2 .

b= 21/3 = e2i/3 F = Q() E= Q(, b) =F(b)

Gal(E/F)= Z3 . H = Gal(E/F) E x3 2 Q F [E :Q] = 6 [F : Q] = 2 |H|= [E :F] = 3

irr(F,b)(x) = irr(Q,b)(x) =x3

2.

H b

b

b

b

2b

H H idE=H (b) =b

2(b) =(b) =() (b) = b = 2b

3(b) =(2(b)) =(2b) =(2) (b) =2 b = 3b= b .

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Galois

E F {1, b , b2} E F

a= a0+a1b+a2b2, aiF .

H (b) =b (a) =(a0+a1b +a2b

2) = a0+ (a1)b + (a22)b2 .

Cauchy G p p |G| gG g p

g1, g2 S5 g1 g2

g1, g2= S5 S5 g1 g2

Gal(E/F) E f(x)F[x] f(x)

f(x) = x5 4x+ 2 E f(x) Q G= Gal(E/Q) f(x) Eisenstein p = 2 Q f(x) = 5x4 4f(x) = 20x3 f(x) x

-2 -1 1 2

-10

-5

5

10

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Galois

Galois

C f(x) a1, a2, a3 a4 a5 a4 = a5 a4 = a+ bi a5 = abi E= Q(a1, . . . , a5) irr(Q,a1)(x) =f(x)

|G

|= [E : Q] = [E: Q(a1)][Q(a1) :Q] = 5[E : Q(a1)].

Cauchy G C

: C C, c +dic di .

|Q= idQ (ai) =ai i= 1, 2, 3

(a

4) =a

5

|E : EE , b E (b) E |E G |E f(x) G S5

Galois f(x) F[x] E/F f(x) FBE Gal(E/F) F B E Gal(E/B ) Gal(E/F)

Gal(E/B ) (b) = b b B Gal(E/F)

F BE B, E F Gal(E/B ) Gal(E/F)

Gal(E/F)/ Gal(E/B)=Gal(B/F).

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Galois

B F B =F(b1, . . . , bn) bi g(x) F[x] Gal(E/F) b B (b) B B bi i= 1, . . . , n (bi)

B (bi)

g(x) (bi) = bj j = 1, . . . , n |B : B B (c) =c, cF

|BGal(B/F).

: Gal(E/F)Gal(B/F), |B . ker

Gal(E/F)

Gal(E/F)/ ker =Im. ker = Gal(E/B ) Im= Gal(B/F)

idB Gal(B/F) ker

{Gal(E/F) : () = idB}={Gal(E/F) : |B = idB}= Gal(E/B ) .

ImGal(B/F) Gal(B/F) Gal(E/F) |B = E F E F B E

b= 21/3 = e2i/3 F = Q() E= Q(, b) G= Gal(E/Q) Q F E |G|= 6 H= Gal(E/F)= Z3 G

b b b

2

b .

F Q H G

G/H=Gal(Q()/Q)=Z2

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Galois

Galois

E F G= Gal(E/F) H G H

EH ={aE : (a) =a,H .

EH FEH E H1H2EH1 EH2 .

E = Q(i) G = Gal(E/Q) ={idE, 1 : i i}= Z2 a +bi= 1(a +bi) =a bib = 0 EG= Q

E= Q(21/3) G= Gal(E/Q) G={idE} EG=E E =Q(, b) = e2i/3 b = 21/3 G = Gal(E/Q)

H =1 1 : b b, EH = Q() Q()EH Q E{1,,b,b,b2, b2} a E

a= a0+a1+a2b+a3b+a4b2 +a5b

2, ai Q.

1(a) =a0+a1+a2b +a3b2 +a4b

22 +a5b23

=a0+a1+a2b +a3b( 1) +a4b2( 1) +a5b2=a0+a1+ (a2

a3)b

a3b+ (a5

a4)b

2

a4b

2 .

1(a) =a a2 = a3= 0 =a4 = a5 .

EH ={a0+a1: ai Q}= Q() .

[E :Q()] = 3 B Q() B E (b)= b EH = E EH = Q()

EGal(E/F) =B

E f(x) F[x]

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Galois

Galois

E/F G= Gal(E/F) G G E/F Galois F E f(x)

F[x]

P F E

Galois E Galois F G= Gal(E/F) P E F G G

Gal(E/) : P G, BGal(E/B )

E : G P, HEH .

[B :F] = [G: Gal(E/B )] [G: H] = [EH :F] EGal(E/B) =B Gal(E/EH) =H B Galois F Gal(E/B) G

EGal(E/B) =B

E Galois F

EGal(E/F) =F .

G= Gal(E/F) |G|= [E :F] G={1 . . . , n} FEG

[E: F] = [E: EG][EG :F],

[E :EG

] =n

[E :EG

] = 1

EG

=F

[E :EG] =n

n= 1 G={idE} E=EG |G|= [E :F] [E :F] = 1 E=F EG =F

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Galois

Galois

n > 1 [E : EG] = m < n {a1, . . . , am} EG E bE EG

b= c1a1+

+cmam= ciai, ciEG . G (ci) = ci i = 1, . . . , m EG

(b) =(

ciai) =

c1(ai) .

m n E

1(a1)x1+ +n(a1)xn = 0

1(am)x1+ +n(am)xn = 0

m < n (y1, . . . , yn)=0 i= 1, . . . , m

y11(ai) + +ynn(ai) =j

yjj(ai) = 0 .

b E bE

y11(b) + +ynn(b) = 0 .

ci EG b =

c1(ai) ci cij(ai) = j(ci)i(ai) = j(ciai) i= 1, . . . , m

j

yjj(ciai) = 0.

i = 1, . . . , m

i

j

yjj(ciai) =j

yjj(c1a1+ +cmam) =m

j=1

yjj(b) = 0.

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Galois

(y1, . . . , yn)= 0 yn= 0 b E

c E n(c)

=1(c) c

= 0 1(c)n(c)

= 0

bE bc y11(bc) + +ynn(bc) = 0 .

1(c)

y11(b) +y22(c)

1(c)2(b) + +ynn(c)

1(c)n(b) = 0 .

y22(c)

1(c)

1(c) 2(b) + +yn

n(c)

1(c)

1(c) n(b) = 0 .

zi = yii(c) 1(c)

1(c)

bE, z21(b) + +znn(b) = 0 , zn= 0

bE, tn(b) = 0, t= 0.

b E n(b) = 0 n E

[E : F]

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Galois

Galois

E=F a1E\F [E :F]

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Galois

E Galois F B g(x)F[x] B E B Galois F B/F BE Galois F (B) =B

Gal(E/F)

Galois E F E F

Galois E Q

E Galois F G= Gal(E/F) f(x)F[x] E[x] X={1, . . . , n} f(x)

= (n n1)(n n1) (2 1) . G () = (2) = 2 EG = F 2 F 2 f(x) i () = F /F B =F() x2 2 F[x] F

2 = [F() :F] = [G: Gal(E/F())]

Gal(E/F()) G .

f(x)

f(x) F[x] deg f(x) = 3 f(x) E f(x) F G= Gal(E/F) S3

2 /F

|G| G=S3 F f(x)

1, 2, 3

= (3 2)(3 1)(2 1) () = G G G G=A3= Z3

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Galois

Galois

E Galois F F BE

Gal(E/F)

Galois

f(x) = x4 2 Q E = Q(b, i) b =21/4 f(x) {b, ib} E Galois Q E Q {1, b , b2, b3,i,ib,ib2, ib3} G= Gal(E/Q) G=D8 G bi

b b b b b ib ib ib ibi i i i i i i i i

idE 1 2 3 4 5 6 7

.

G G 4 G G

H1= ={0, 4, 2= 24, 6 = 34}

H2={0, 2, 1, 3}

H3={0, 2, 5, 7}

={idE, 3}

={idE, 1}

={idE, 2}

={idE, 5} ={idE, 7}

G

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Galois

(1)

H2

H1 H3

G

G

E

y= a0+a1b+a2b2 +a3b

3 +a4i +a5ib+a6ib2 +a7ib

3,

ai Q

5(y) =a0+a1ib a2b2 ia3b3 a4i +a5b+a6ib2 a7b3

5(y) = y

a1 = a5, a2 = 0, a3=a7, a4= 0,

y= a0+a1b(1 +i) +a3b3(1 i) +a6ib2.

E = Q(b(1 +i), b3(1 i), ib2), (b(1 +i))2 = 2ib2 (b(1 +i))3 =2b3(1 i)

E = Q(b(1 +i)).

E [G: ] = 4 2(i) =i 2(b

2) =2(b)2 = (b)2 =b2

Q(b2, i)E

Q Q(i) Q(i, b2

) E

[Q(i, b2) :Q] = 4 E =Q(i, b2) E

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Galois

E= Q(b, i)

Q(b)

Q(bi)

Q(b2, i) Q(b(1 +i))

Q(b(1 i))

Q(b2)

Q(i) Q(b2i)

Q

B2 = Q(b2) x22 Q Gal(E/B2)=H2

B1 = Q(i) x2+1 Q Gal(E/B1)

=

H1

B3 = Q(b2i) x2 + 2 Q Gal(E/B3)=H3

B4 = Q(b2, i) (x2 b)(x2 + 1) Q Gal(E/B4)=< 2 >

C=a n

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Galois

g(C) C |C| = m |g(C)| =(m) C1=C2 g(C1) g(C2) =

G=

Cg(C)

C G n G = Zn

|Zn|=C

|g(C)|

C Zn

n=

d(d) ,

d n d Zn n

n=d/n

1dn

(d) .

G n d n d

G d|n d d n d

n= C |g(C)| d|n1dn

(d) = n .

d n G d d= n G

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Galois

k

G

(K {0}, )

n

d|n

C G d Lagrangecd = 1, cC G d d + 1 x G xd = 1 xd 1 d d G d n G

k k= (k {0}, ) k = Zp(a) p a

k k

2 Z11

f(x) =x2 2 Z5 Z5 k f(x) Z5 a k f(x) f(x) a k = Z5(a) |K|= 25 k k+la k, l Z5 a a2 = 2 b= 2 + a b k= Z5(b)

k pn a k a n

k = Zp(a) [k : Zp] f(x) = irr(Zp,a)(x)

|k|=|Zp|[k:Zp]

deg f(x) =n

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Galois

Galois Gal(k/Zp)

k pn

Gal(k/Zp)=Zn G = Gal(

k/

Zp) a

k f(x) =

irr(Zp,a)(x) k =Zp(a) deg f(x) =n k n f(x) k f(x)

|G|= [k : Zp] =n f(x) G (a) K a (ai) = (a)i (a) f(x) deg f(x) =n f(x) n (a) n

|G

| n .

|G| = n G G

: k k, bbp G k Zp Zp p 1

c Zp{0} cp1 = 1 cp =c c Zp Zp

k k G , 2, . . . , n G i < n i = idk b k

i(b) =bbpi =bbpi b= 0 .

bk xpi x |k| = pn deg xp

i x = pi < pn || n G

|G| n

|G|=n G= G n (Zn, +)

|k|= pn

p

Zp

k: k k, bbp

Gal(k/Zp) Frobenius

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Galois

Zn ={a Zn : (a, n) = 1} . Zn |Zn|= (n) (8) = 4 Z8 ={1, 3, 5, 7} 3 5 7 Z8 Z

8

p Zp

Zp Zp

k n k n f(x) =xn 1 k[x] E f(x) k f(x) E {0} n n k

n k

e2i/n n Q

F F = Z2(a) a F 3 Z2

k k n f(x) =xn 1 k[x] L f(x) Gal(L/k) Zn

G = Gal(L/k) n

L L = k() n L G () = i i {1, . . . , n}

=|: , i

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Galois

i Im =i (i, n) = 1

: G

Zn,

i, () =i .

ker ={G : i= 1}={G : () = }={idL}

= e2i/p p irr(Q,)(x) = p(x) =x

p1 + +x + 1

| Gal(Q()/Q)|= p 1.

Gal(Q()/Q) Zp |Z

p|= p 1

Gal(Q()/Q)=Zp .

Zp Gal(Q()/Q)

f(x) =xna k[x] k n k n L f(x) G = Gal(L/k)

f(x)

b,b, , bn1

b

f(x)

L= k(b) G (b) =bi i (b) f(x) i

: G Zn, i .

, G () = ()(b) =(b) ,

G Zn f(x) k[x] irr(k,b)(x) = f(x)

|G| = [L :k] = deg f(x) =n G Zn G Zn

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Galois

Galois

f(x) f(x)

f(x) = g(x)h(x), g(x), h(x) k[x], (g(x), h(x)) = 1 . g(b) = 0 h(bi) = 0 i G (b) =bi g(b) = 0 irr(k,b)(x) g(x)

irr(k,(b))(x) = irr(k,b)(x).

irr(k,b)(x) h(x) irr(k,b)(x) (g(x), h(x))

k k n k n L f(x) = xna k[x] Gal(L/k) (Zn, +) Gal(L/k)

Zn

f(x)

B =Q() E=B() b= 21/3 E x3 2B[x] Gal(E/B)= Z3

p k p p ak xp a k[x] xp a k[x]

L xp a Gal(L/k) Zp Gal(L/k)={0} Gal(L/k)= Zp L= k xp a

Q[x] 4

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Galois

G G = G0 G1 . . . Gn ={e} Gi+1 Gi Gi/Gi+1

G H G

G H G G/H

f(x)

Q[x] E f(x)

f(x)

Gal(E/Q)

f(x) F[x] E f(x) f(x) B/E a1, . . . , at B B = F(a1, . . . , at) Bi = Bi1(ai) amii Bi1 mi N

f(x) =x3 +q(x) +r Q[x] f(x) y+z, 3y+

23z,

23y+3z

3= e2i/3 ,

y= (1

2

r+r2 +4q327

) 1

3

z= (1

2

r r2 +4q327

) 1

3 .

f(x)

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Galois

Galois

f(x) L =Q(y+z, 3y+om23z,

23y+3z) f(x)

a1= e2i/12, a2= r

2 +4q3

27, a3 = y, a4 = z

B = Q(a1, . . . , a4) LB B0= Q B1= B0(a1) B2= B1(a2) B3= B2(a3) B4 = B3(z) =L

3 =i 4 =3

a121 Q a22= r2 + 4q

3

27 Q B1

a33= 12(r+

r2 + 4q3

27)B2

a34=

1

2

(

rr2 + 4q

3

27

)

B2

B3

ai i= 1, . . . , 4

f(x)Q[x] E f(x) B= Q(a1, . . . , at) B Galois Q a1 mi Gal(B/E) Gal(B/Q)

Gal(E/Q)=Gal(B/Q)/ Gal(B/E) . Gal(B/Q) Galois

Q =B0B1 Bt1Bt =B Bi = Bi1(ai) i = 1, . . . , t Gal(Bi/Bi1) i= 1, . . . , t Gal(Bi/Q)

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Galois

[Q(c) : Q] = 2 c

Q =K0K1 Kt = Q(c)

[Ki+1: Ki] = 2 i= 1, . . . , t 1 c [Q(c) : Q] = 2 irr(Q,c)(x) c (a d)/b

d c

d

d

c c Q [Q(c) : Q] = 2l l N c n ui i u1 Q Q u1

Q(

v1) v1 Q

[Q(u1) : Q] = [Q(v1) : Q] [Q(u1) : Q] = 1 v1 Q [Q(u1) : Q] = 2

v1 / Q

c u2 Q(u1, u2) [Q(u1, u2) :Q(u1)]

Q Q(u1) Q(u1, u2) Q(u1, u2, . . . , un)

u2i Q(u1, u2, . . . , ui1) [Q(u1, u2, . . . , ui) : Q(u1, u2, . . . , ui1)] 2in c Q(u1, u2, . . . , un) [Q(u1, u2, . . . , un) :

Q]

2

n

c

[Q(c) : Q] = 2l

cR [Q(c) :Q] = 2l c

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Galois

Galois

60

60 20 70

cos(20)

cos(3) = 4 cos3() 3 cos() cos(60) = 1/2 cos(20) 8x3 6x 1 Q Q[x] [Q(cos(20)) : Q] = 3 cos(20) 60

a= 32 a x32 [Q(a) : Q] = 3 a

Q

p

p

e2i/p

G |G| = 22m G

GG= Gt Gt1 {e}

[Gi : Gi1] = 2, i= 1, . . . , t

Gauss

Gauss p p

p= 22m

+ 1, m N .

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Galois

p = e2i/p z [Q(z) : Q]

irr(Q,)=xp1 + +x+ 1

p 1 = 2s s s k s = k

p= 2s + 1 = (2)k + 1

2 + 1 p p 1 = 22m m0 p

p= 22m

+ 1 m N,

| Gal(Q()/Q)|= [Q(z) : Q] =p 1 = 22m . Gal(Q()/Q)

G= Gt Gt1 {e}.

Galois

Q =K0K1 Kt = Q() [Ki+1: Ki] = 2 i= 1, . . . , t 1

22m

+ 1 Fer-mat n

n

n= 2s

p1 pr, s p1, . . . , pr Fermat

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Galois

Galois

Galois Sylow

G G 2m |G| = 2mk (m, k) = 1 1nm G H |H| = 2n

f(x) R[x] a, b R f(a)>0 f(b)< 0 f(x)

a R a0 rR r2 =a

f(x) =x2 a R[x] f(1 + a) = 1 + a2 + a f(1 +a) >0 f(0) =a

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Galois

f(x) R[x] f(x) R[x]

c0 + c1x+

+cnxn

c0/cn + c1/cnx+ + xn f(x) =a0+a1x + +xn R[x] deg f(x)

t= 1 +i=n1i=0

|ai|.

t 1 =i=n1i=0

|ai| |ai| t 1, i= 1, . . . , n 1

|a0+a1t+ +an1tn1| (t 1)(1 +t + +tn1) = tn 1< tn .

f(t) = (a0+a1t1+ +an1tn1) +tn >0. n (t)n = (1)tn 1 a L/R irr(R,a)(x)

[L: R] = [L: R(a)][R(a) : R]

C[x]

f(x)C[x] f(x) f(x) C f(x)f(x)

f(x)f(x) R[x] z f(x)