16 B

16 B

(1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i

1 1. 1.1.. 1. (1) R + 2 (a) (R, +) (i) (a + b) + c = a + (b + c) ( a, b, c R) (ii) 0 R s.t. a + 0 = 0 + a = a ( a R) 0 (iii) a R, b R s.t. a + b = b + a = 0 b a a (iv) a + b = b + a ( a, b R) (b) (R, ) 1 ( ) (R, +) 0 (v) (a b) c = a (b c) ( a, b, c R) (vi) 1 R (1 0) s.t. a 1 = 1 a = a ( a R) (vii) a b = b a ( a, b R) (c) (a + b) c = a c + b c ( a, b, c R) (2) R R a, b R (viii) ab = 0 a = 0 b = 0 (3) R R (ix) a 0 ( a R) b R s.t. ab = ba = 1 R a b a a 1 ( (vi) ) 2. ab = 0, a 0 a 1 b = a 1 ab = a 1 0 = 0 3. (1) N = {0, 1, 2, } (2) Z = {0, ±1, ±2, } Z (3) Q R C 4. K = {a + b 2 R a, b Q} ( 2) 2 = 2 K K 2 a, b 0 a ± 2b 0 a 2 2b 2 0 (a + b 2) 1 = a b 2 a 2 2b 2 K 0

2 5. R 0 a 0 b, b a b = 1 b = (b a)b = b (ab) = b 1 = b 1.2.. 2 6. (1) L L K L L K (2) K L L K L/K (3) L/K K L L/K 7. (1) Q R C R C/Q (2) K 4 Q K R K R/Q 8. L K L L 0 a, b K a b, ab 1 K (b 0) K L 9. L (1) M L/K M K (2) M/K L/M L/K 8 10. L/K X L K(X) = M M : X L X K X = {x 1, x 2,, x n } K(X) = K(x 1, x 2,, x n )

3 11. L/K {M λ } λ Λ L/K M = λ Λ M λ 12. K 4 R K = Q( 2) K R/Q 2 K K Q( 2) Q( 2) Q 2 a + b 2 (a, b Q) R K Q( 2) 1.3.. 13. L/K L K L K L 14. L/K (1) L K L/K L K L/K [L : K] (2) L K L/K L/K [L : K] = 15. (1) [R : Q] =, [C : Q] =. (2) [C : R] = 2. (3) [Q( 2) : Q] = 2. 16. L/K M (i) L/K (ii) M/K L/M [L : K] = [M : K][L : M] ( = )

4 (i) (ii) (ii) (i) M K m 1, m 2,, m e L M l 1, l 2,, l d l i m j (1 i d, 1 j e) L K a ij l i m j = 0 (a ij K) i,j l 1, l 2,, l d M L i a ij m j = 0 j m 1, m 2,, m e K M a ij = 0 ( i, j) L K l i m j (i) (ii) [L : K] = de = [M : K][L : M] 17. L/K K L 18. L/K L = K [L : K] = 1 L K K 1 19. L/K α 1,, α n L L = K(α 1,, α n ) [L : K] [L : K] = 1 [L : K] > 1 α L \ K L = K(α) L K(α) [L : K(α)] = [L : K]/[K(α) : K] < [L : K] β 1,, β m L L = K(α)(β 1,, β m ) = K(α, β 1,, β m ) 20. Q( 2, 3) Q R Q( 2, 3) = {a + b 2 + c 3 + d 6 a, b, c, d Q} [Q( 2, 3) : Q] = 4 Q( 2) Q( 2)/Q 16 Q( 2, 3)/Q( 2) [Q( 2, 3) : Q( 2)] = [Q( 2, 3) : Q]/[Q( 2) : Q] = 4/2 = 2

5 L = {a + b 2 + c 3 + d 6 a, b, c, d Q} (a + b 2 + c 3 + d 6) + (s + t 2 + u 3 + v 6) = (a + s) + (b + t) 2 + (c + u) 3 + (d + v) 6 (a + b 2 + c 3 + d 6)(s + t 2 + u 3 + v 6) = (as + bt + cu + dv) + (at + bs + 3cv + 3du) 2 +(au + 2bv + cs + 2dt) 3 + (av + bu + ct + ds) 6 L R L 1, 2, 3, 6 L Q a + b 2 + c 3 + d 6 = 0 a = b = c = d = 0 a, b, c, d 3 a + b 2 = c 3 d 6 a 2 + 2ab 2 + 2b 2 = 3c 2 + 6cd 2 + 6d 2 2 { a 2 + 2b 2 = 3c 2 + 6d 2 2ab = 6cd (i) c = d = 0 a = b = 0 (ii) c 0, d = 0 a, b 0 a = 0 2b 2 = 3c 2 3 3 b = 0 c = 0, d 0 (iii) cd 0 2 a, b 3 3 2 c, d 3 c, d 3 a, b, c, d 3 L Q 4 0 4 L (a + b 2 + c 3 + d 6)(a + b 2 c 3 d 6)(a b 2 + c 3 d 6)(a b 2 c 3 + d 6) 0 L 0 L L = Q( 2, 3) Q( 2, 3) Q 4 21. Q( 2, 3) = Q( 2 + 3) Q( 2, 3) Q( 2 + 3) 2, 3 Q( 2 + 3) Q( 2, 3) Q( 2 + 3) α = 2 + 3 α 2 2 2α + 2 = (α 2) 2 = ( 3) 2 = 3 α 2 1 2 = 2α, α 2 + 1 3 = 2α Q( 2 + 3) 1. d = 1 d = ±p 1 p 2 p r (p i ) (1) d (= ) (2) F = Q( d) Q 2

6 2. Cauchy 3. (1) (2) 15

7 2. 2 2.1.. 22. R S (1) ϕ : R S (i) ϕ(a + b) = ϕ(a) + ϕ(b) ( a, b R) (ii) ϕ(ab) = ϕ(a)ϕ(b) ( a, b R) (iii) ϕ(1) = 1 (2) ϕ : R S ψ : S R ψ ϕ = id R ϕ ψ = id S (3) R S ϕ : R S A A A A 23. ϕ : R S (1) ϕ( a) = ϕ(a) (2) ϕ(a 1 ) = ϕ(a) 1 24. 0 0 25. ϕ : R S (i) ϕ (ii) ϕ (ii) (i) ϕ ϕ 1 : S R a, b S ϕ(ϕ 1 (a + b)) = a + b = ϕ(ϕ 1 (a)) + ϕ(ϕ 1 (b)) = ϕ(ϕ 1 (a) + ϕ 1 (b)) ϕ ϕ 1 (a + b) = ϕ 1 (a) + ϕ 1 (b) ϕ 1

8 26. 2 K L 2.2.. - 27. R K ι : R K (K, ι) ( ) (K, ι) R R K K ( ) F θ : R F α : K F R θ ι F α K (K, ι) ( ) R (R \ {0}) (a, b) (c, d) ad bc = 0 ( ) K R (R \ {0}) K = R (R \ {0})/ (a, b) a b K a b + c a c d b d = = ad + bc ac bd bd K 0 1 0 1 1 1 ι : R K 1 ι(a) = a 1 ι ι ι(a) = 0 = 0 1 a = a 1 = 0 1 = 0 ι ker(ι) 0 ι ( ) : α : K F α( a b ) = θ(a) θ(b) α α( a b ) = α(ι(a)) α(ι(b)) = θ(a) θ(b) α θ α ( )

9 (K, ι) : (K, ι ) (K, ι) (K, ι ) 2 R ι ι K α (K, ι) (K, ι ) K R ι ι K β K β α = id K α β = id K 28. K K 29. (1) Z Q (2) K K[x] K(x) 30. L/K α L K[α] K(α) 2.3.. 31. R (1) R I (i) x, y I x + y I. (ii) a R, x I ax I. R (2) R p ab p a p b p (3) R m R I m I I = m - 32. I R (R, π : R R) ( ) ( ) S ϕ : R S ϕ(i) = {0} ψ : R S π R R ϕ ψ S π : R R (R, π : R R) R I π R R/I

10 R/I R a b a b I R/I R R/I a R a R/I a + b = a + b ab = ab well-defined 27 33. I R (1) I R/I. (2) I R/I. (2) : I x R x 0 R/I J = {ax + y a R, y I} J R I x I x = x + 0 J I J = R a R y I ax + y = 1 ax = 1 x R/I : R/I I R J J = I I x J \ I R/I ax = 1 a R 1 ax + I J J J = I 34. Zorn 35. I R I 2.4. p. Z 36. R (principal ideal domain, P.I.D. ) a R R (a)

11 37. a 0 b q, r b = qa + r (0 r < a ) 38. (1) Z P.I.D. (2) {Z } { } (a) a (a > 0). Z (0) - 39. p Z/(p) p F p p 2.5.. ( ) 40. ϕ : R S (1) ϕ ker(ϕ) = {a R ϕ(a) = 0} R (2) ϕ im(ϕ)(= ϕ(r)) S (3) ϕ R/ker(ϕ) im(ϕ) a ϕ(a) 41. R Z R ϕ : Z R 1 + 1 + + 1 (n ) if n > 0 n 0 if n = 0 ϕ( n) if n < 0 m, n mn = 1 + 1 + + 1 (mn ) ϕ Z 1 Z R ϕ

12-42. K ϕ : Z K ker(ϕ) ker(ϕ) = (0) K 0 p ker(ϕ) = (p) K p K char(k) K K im(ϕ) 33 40 ker(ϕ) 38 27 32-43. K (1) char(k) = 0 Q K. (2) char(k) = p > 0 F p K. K Q F p K 4. Zorn 35 5. 37, 38 6. ( )

13 3. 3.1. P.I.D. K K K[x] ( ) P.I.D. N N N = N { } a + ( ) = + a = ( a N) < a 44. a, b, c N (1) N (a + b) + c = a + (b + c) (2) N (i) a a. (ii) a b b a a = b. (iii) a b b c a c. N 45. K f(x) K[x] N deg(f(x)) { n if f(x) = a0 x deg(f(x)) = n + + a n (a i K, a 0 0) if f(x) = 0 46. f(x), g(x) K[x] (1) deg(f(x)) = f(x) = 0. (2) deg(f(x) + g(x)) inf{deg(f(x)), deg(g(x))}. (3) deg(f(x)g(x)) = deg(f(x)) + deg(g(x)). 47. f(x) K[x], f(x) 0 g(x) K[x] q(x), r(x) K[x] g(x) = q(x)f(x) + r(x) (deg(r(x)) < deg(f(x)))

14 deg(g(x)) deg(g(x)) < deg(f(x)) q(x) = 0, r(x) = g(x) deg(g(x)) deg(f(x)) f(x) g(x) m n (m n) a, b a, b 0 [ deg g(x) b ] a xn m f(x) < deg(g(x)) g(x) b a xn m f(x) = s(x)f(x) + t(x) (deg(t(x)) < deg(f(x))) q(x) = b a xn m + s(x), r(x) = t(x) 48. K[x] P.I.D. I K[x] ( (0) ) n = inf{deg(f(x)) f(x) I, f(x) 0} 46 (1) 44 n f(x) I deg(f(x)) = n I = (f(x)) I (f(x)) g(x) I 47 g(x) = q(x)f(x) + r(x) (deg(r(x)) < deg(f(x))) q(x), r(x) K[x] I r(x) = g(x) q(x)f(x) I f(x) deg(r(x)) = r(x) = 0 g(x) = q(x)f(x) I (f(x)) 49. f(x), g(x) K[x] (f(x) 0) f(x) g(x) g(x) (f(x)) f(x) g(x) 3.2.. 50. K[x] K[x] K[x] = K K K[x] f(x)g(x) = 1 46 (3) deg(f(x)) + deg(g(x)) = deg(f(x)g(x)) = deg(1) = 0 f(x) g(x) 0 K[x] = K

15 51. (1) f(x) K[x] 0 f(x) (monic) f(x) 1 (2) p(x) K[x] 0 p(x) (irreducible) p(x) = f(x)g(x) f(x) g(x) K[x] p(x) 52. K[x] 1 53. (1) f(x) = ax 2 + bx + c R[x] (a 0) 2 D = b 2 4ac f(x) D < 0 (2) g(x) R[x] g(x) 54. (1) (2) {K[x] } {K[x] } \ {(0)} f(x) (f(x)). {K[x] } {K[x] } p(x) (p(x)). K[x] (0) (1) K[x] P.I.D. 50 (2) K[x] (1) 2 1 K[x] (p(x)) p(x) 2 p(x) (p(x)) 1 : p(x) = f(x)g(x) (p(x)) f(x) (p(x)) g(x) (p(x)) f(x) (p(x)) h(x) K[x] f(x) = h(x)p(x) f(x) = h(x)p(x) = f(x)g(x)h(x) K[x] g(x)h(x) = 1 g(x) p(x) g(x) (p(x)) 2 : (p(x)) K[x] P.I.D. (p(x)) (f(x)) K[x] f(x) p(x) (f(x)) g(x) K[x] p(x) = f(x)g(x) p(x) g(x) (f(x)) = (p(x)) (p(x))

16-55. f(x) K[x] 0 a K p 1 (x), p 2 (x),, p d (x) f(x) = ap 1 (x)p 2 (x) p d (x) f(x) p i (x) f(x) deg(f(x)) f(x) 1 2 g(x) h(x) f(x) = g(x)h(x) 46 (3) g(x) h(x) f(x) f(x) f(x) = bq 1 (x)q 2 (x) q e (x) (b K, q i (x) ) a = b 54 (2) (p 1 (x)) q 1 (x)q 2 (x) q e (x) (f(x)) (p 1 (x)) j (q j (x)) (p i (x)) (q j (x)) 54 (2) (q j (x)) = (p i (x)) q j (x) = p i (x) 3.3.. 56. f(x) K[x] 0 α K f(x) f(α) = 0 57. f(x) K[x] 0 α K (i) α f(x) (ii) x α f(x) 1 47 f(x) = (x α)q(x) + r (r K) q(x) K[x] f(α) = 0 r = 0 58. f(x) K[x] 0 d f(x) d 55 f(x) 46 d 57 1

17 7. 53 8. µ K µ

18 4. L/K 4.1.. 59. α L (1) α K 0 K f(x) α K (2) α K K 60. (1) 2 R Q (2) π R Q 61. (1) L/K L (2) L/K 62. (1) Q( 2)/Q α = a + b 2 Q( 2) (a, b Q) f(x) = x 2 2ax + a 2 2b 2 Q[x] α f(x) (2) Q(π)/Q (3) R/Q C/Q (4) C/R 63. L/K n α L (α 0) [L : K] = n 1, α, α 2,, α n K 1 a 0 α n + a 1 α n 1 + + a n 1 + a n = 0 (a 0,, a n ) (0,, 0) 0 K α 64. [Q( 2, 3) : Q] = 4 ( 20) Q( 2, 3)/Q

19 4.2.. - 65. α L K α K p(x) α f(x) p(x) f(x) p(x) α K p α,k (x) α I K[x] α K I ( 54 (1)) 66. α L K α K p α,k (x) p α,k (x) K p α,k (x) = q 1 (x)q 2 (x) q d (x) p α,k (α) = 0 i q i (α) = 0 p α,k (x) q i (x) p α,k (x) = q i (x) p α,k (x) 67. α L K M L/K α M M[x] p α,m (x) p α,k (x) 4.3.. 68. L/K α L L = K(α) 69. K(α) K K ϕ : K[x] K(α) an x n a n α n K[x]/(p α,k (x)) = K(α) [K(α) : K] = deg(p α,k (x)) ker(ϕ) α p α,k (x) ( 65, 66) 54 im(ϕ) = K[x]/(p α,k (x)) im(ϕ) α K im(ϕ) = K(α) n = deg(p α,k (x)) K[x]/(p α,k (x)) 1, x,, x n 1 K [K(α) : K] = n

20 70. α L (i) α K (ii) [K(α) : K] <. (i) (ii) 69 (ii) (i) 63 67 71. α L K M L/K [M(α) : M] [K(α) : K] 72. L/K M (i) L/K (ii) L/M M/K (i) (ii) 67 (ii) (i) : α L α M p α,m (x) = x n + a 1 x n 1 + + a n M[x] K(α) K(a 1,, a n, α) 63 K(a 1,, a n, α) K a 1, a 2,, a n K 72 [K(a 1,, a n, α) : K] = [K(a 1 ) : K][K(a 1, a 2 ) : K(a 1 )] [K(a 1,, a n, α) : K(a 1,, a n )] [K(a 1 ) : K][K(a 2 ) : K] [K(a n ) : K]deg(p α,m (x)) < 73. Q 2 2 F 2 d = 1 d = ±p 1 p 2 p r (p i ) F = Q( d) ( 1 ) 74. α = 2 + 3 Q p α,q (x) = x 4 6x 2 + 1 Q( 2, 3) = Q( 2 + 3) ( 21) [Q( 2 + 3) : Q] = 4 ( 21) 69 4 p α,q (x) = x 4 6x 2 + 1 α 2 = 3 α 2 2 2α + 2 = 3 α 2 1 = 2 2α α 4 2α 2 + 1 = 8α 2 α 4 6α 2 + 1 = 0

21 75. L = K(α, β) K [K(α) : K] [K(β) : K] [L : K] = [K(α) : K][K(β) : K] 16 [L : K] [K(α) : K] [K(β) : K] 71 [K(α) : K][K(β) : K] 9. π e Q 10. L/K M L K M L/K 11. L/K L K e 1, e 2,, e d (d = [L : K]) L K α L K l α : L L l α (v) = αv l α (e 1, e 2,, e d ) = (e 1, e 2,, e d )A α L Mat(d, K) α A α L K Mat(d, K) K d (1) α, β L A α+β = A α + A β A αβ = A α A β (2) a K A a = ae d E d d 12. L/K L K (α A α ) T L/K : L K α trace(a α ) N L/K : L K α det(a α ) L K (1) α, β L (2) a K, α L T L/K (α + β) = T L/K (α) + T L/K (β) N L/K (αβ) = N L/K (α)n L/K (β) T L/K (a) = [L : K]a, N L/K (a) = a [L:K] (3) M L/K T L/K (aα) = at L/K (α) T L/K = T M/K T L/M N L/K = N M/K N L/M 13. L/K L K (α A α ) (1) A α p α,k (x) (2) A α p α,k (x) [L:K(α)]

22 5. 5.1.. 76. 0 f(x) = a n x n + a n 1 x n 1 + c + a 0 Z[x] Z (a 0, a 1,, a n ) = Z 77. 2 f(x), g(x) Z[x] f(x) = a 0 x m + a 1 x m 1 + + a m (a 0, a 1,, a m ) = (1) g(x) = b 0 x n + b 1 x n 1 + + b n (b 0, b 1,, b n ) = (1) f(x)g(x) p f(x)g(x) p p f(x) g(x) k (0 k m) l (0 l m) p a i (0 i k 1), p a k p b j (0 j l 1), p b l f(x)g(x) x m+n k l a 0 b k+l + a 1 b k+l 1 + + a k 1 b l+1 + a k b l + a k+1 b l 1 + + a k+l 1 b 1 + a k+l b 0 i > m a i = 0 b j a k b l k, l p a k b l p f(x)g(x) 78. 1 f(x) 1 f(x) = g(x)h(x), deg(h(x)), deg(g(x)) 1 a ag(x) b bh(x) abf(x) = ag(x) bh(x) 77 abf(x) f(x) ab = 1 f(x) 2 1

23 5.2.. 79. p f(x) = x n + a 1 x n 1 + + a n (n 1) p a i (1 i n), p 2 a n f(x) f(x) 2 1 f(x) = (x l + b 1 x l 1 + + b l )(x m + c 1 x m 1 + + c m ), m + l = n, l, m 1 p a n p 2 a n p b l p c m p c m k (0 k l 1) p b k k 77 x l k a k+m p k + m > 0 p a i (1 i n) p b l f(x) 78 80. (1) n 1 f(x) = x n 2 [Q(2 1 n ) : Q] = n p = 2 (2) f(x) = x 4 + x 3 + x 2 + x + 1 ζ 5 1 5 [Q(ζ 5 ) : Q] = 4 x = y + 1 f(x) = x5 1 x 1 = (y + 1)5 1 = y 4 + 5y 3 + 10y 2 + 10y + 5 y y x y 1 f(x) 81. ω 1 3 ω 2 + ω + 1 = 0 [Q(2 1 3, ω) : Q] = 6 Q(2 1 3, ω) = Q(2 1 3 + ω) 80 (1) [Q(2 1 3 ) : Q] = 3 80 (2) [Q(ω) : Q] = 2 75 6 21 5.3.. p p Z Z/(p) = F p Z[x] F p [x] a a f(x) = a 0 x n + a 1 x n 1 + + a n f(x) = a 0 x n + a 1 x n 1 + + a n p

24 82. f(x) f(x) f(x) f(x) g(x), h(x) (deg(g(x)), deg(g(x)) 1) f(x) = g(x)h(x) f(x) = g(x)h(x) g(x) h(x) 1 f(x) 78 83. (1) f(x) = x 2 + x + 1 2 F 2 [x] f(x) 2 2 1 f(0) = 0 2 + 0 + 1 = 1 0 f(1) = 1 2 + 1 + 1 = 1 0 f(x) 1 82 f(x) (2) g(x) = x 4 + x + 1 2 F 2 [x] g(x) 4 1 2 2 g(0) = 0 4 + 0 + 1 = 1 0 g(1) = 1 4 + 1 + 1 = 1 0 g(x) 1 F 2 [x] 2 x 2, x 2 + x = (x + 1)x, x 2 + 1 = (x + 1) 2, x 2 + x + 1 x 2 + x + 1 (x 2 + x + 1) 2 = x 4 + x 2 + 1 g(x) g(x) 2 82 g(x) 14. (1) p f(x) = x p 1 + x p 2 + + x + 1 (2) g(x) = x 6 + x 3 + 1 15. (1) f(x) = x 3 x + 2 (2) g(x) = x 4 + x 2 + x + 1

25 6. 6.1. C. 84. K 1 K K 85. K (i) K (ii) 1 K f(x) a, α 1,, α n K f(x) = a(x α 1 ) (x α n ) (iii) K α K α K (iv) L/K L = K (v) L/K L = K C (1799 ) 86. C 6.2.. 2 87. K K K K K Zorn 88. K (1) K (2) K 1, K 2 K K K 1 = K2 89. Q C Q Q Q

26 Q : α, β Q [Q(α, β) : Q] [Q(α) : Q][Q(β) : Q] < 70 Q(α, β) Q α + β, αβ, α 1 (α 0) Q Q : f(x) = x n + a 1 x n 1 + + a n Q[x] (n 1) α C [Q(α) : Q] [Q(a 1,, a n, α) : Q] [Q(a 1,, a n ) : Q][Q(a 1,, a n, α) : Q(a 1,, a n )] < 70 α Q 16. 17. 88

27 7. K K K K 7.1.. 2 90. α K α K p α,k (x) K 91. d = 1 ±p 1 p r (p 1,, p r ) a + b d Q( d) (a, b Q) Q a + b d a b d a + b d p a+b d,q = x 2 2ax + a 2 b 2 d = (x a b d)(x a + b d) 92. α K β α K ϕ : K(α) K(β) ϕ(α) = β p α,k (x) = p β,k (x) K K(α) α = K[x]/(p α,k (x)) K(β) x β α β = 93. ω ω 2 + ω + 1 = 0 2 1 3 Q(2 1 3 ) Q 2 1 3, 2 1 3 ω 2 1 3 ω 2 ω 3 = 1 2 1 3 p 2 1 3,Q = x 3 2 = (x 2 1 3 )(x 2 1 3 ω)(x 2 1 3 ω 2 ) 2 1 3 Q(2 1 3 ) R 2 1 3 ω Q(2 1 3 ) R Q(2 1 3 ) Q(2 1 3 ω) Q 3 ( 80) Q Q(2 1 3 ) Q(2 1 3 ω) = Q Q(2 1 3 ) = Q(2 1 3 ω)

28 7.2.. 94. L/K K ι : L K K L K Hom K (L, K) K 95. α K K(α)/K (1) ι Hom K (K(α), K) ι(α) α K ι K(α) K(ι(α)) (2) Hom K (K(α), K) {β K β α } ι ι(α) (1) p α,k (x) α K ι K p α,k (ι(α)) = ι(p α,k (α)) = ι(0) = 0 ι(α) α ι(k(α)) K(ι(α)) K ι(k(α)) = K(ι(α)) K (2) 92 (1) 96. α K Hom K (K(α), K) [K(α) : K] p α,k (x) 97. f(x) K[x] 1 K f(x) = a(x α 1 )(x α 2 ) (x α n ) f(x) K α 1,, α n 2 98. p K = F p (t) K L L = K(t 1 p ) = Fp (t 1 p ) Hom K (L, K) = 1 t 1 p K L p t 1 p,k (x) = x p t = (x t 1 p ) p K t 1 1 p L K t p L K

29 99. L/K Hom K (L, K) < M L/K Hom K (L, K) = Hom K (M, K) Hom M (L, K) 19 L = K(α 1,, α n ) 95 Hom K (L, K) < p : Hom K (L, K) Hom K (M, K) (1) - (3) (1) p (2) σ Hom K (L, K) ι ι M p 1 ({p(σ)}) = Hom σ(m) (σ(l), K) (σ L ) (3) σ Hom K (L, K) σ : K K σ L = σ K ( 88 (2) ) Hom M (L, K) Hom σ(m) (σ(l), K) ι σ 1 ι σ 1 σ(l) (1) : L = M(α) p α,m (x) = x n + a 1 x n 1 + + a n β p ι α,m (x) = x n + ι(a 1 )x n 1 + + ι(a n ) p ι α,m (x) K L = M(α) ai α i = M[x]/(p α,m (x)) K ai x i ι(a i )β i L K L/M K (2) (3) well-defined 100. L/K Hom K (L, K) [L : K] 19 L/K 16 96 99 7.3.. - 101. (1) L L L L L Aut(L) (2) L/K L K L L K L K K Aut(L/K)

30 102. L/K L K Aut(L/K) = {ι Hom K (L, K) ι(l) L} ι L K K ι(l) L K ι(l) = L ι L K 103. L/K Aut(L/K) [L : K] 100 104. α K K(α)/K Aut(K(α)/K) {β K(α) β α } ι ι(α) 105. (1) Aut(Q(2 1 3 )/Q) = 1. 93 Q(2 1 3 ) Q(2 1 3 ω) Q(2 1 3 ) Q(2 1 3 ω 2 ) (2) Aut(Q(2 1 4 )/Q) = 2. Aut(Q(2 1 4 )/Q) = Z/2Z. 80 (1) 2 1 4 p2 1 4,Q (x) = x 4 2 ±2 1 4, ±2 1 4 1 Q(2 1 4 ) ±2 1 4 104 2 (3) Aut(F p (t 1 p )/Fp (t)) = 1. 98 18. L/K ι Hom K (L,K) a ι ι = 0 (a ι K) = a ι = 0 ( ι) 19. (1) n Aut(Q(2 1 n )/Q) (2) Aut(Q( 2 + 5/Q) 20. Aut(R/Q)

31 8. K K K K 8.1.. 106. f(x) K[x] 0 (1) f(x) K f(x) (2) f(x) 107. p F p [x] x p 1 ( ) p p i x p 1 = (x 1) p (0 < i < p) - 108. K d : K[x] K[x] dx f(x) = a n x n f (x) = na n x n 1 d dx (f(x)g(x)) = ( d d f(x))g(x) + f(x)( dx dx g(x)) K 109. f(x) K[x] 0 (i) f(x) (ii) K[x] (f(x), f (x)) = K[x] f(x) 0 f(x) K[x] 1 K f(x) = a(x α 1 ) e1 (x α 2 ) e2 (x α m ) em α 1,, α m e 1,, e m 1 (i) (ii) : f(x) e 1 = e m = 1 f (x) = i a(x α 1 ) (x α i 1 )(x α i+1 ) (x α m )

32 f (α i ) = j i (α i α j ) 0 f(x) f (x) K (f(x), f (x)) = K[x] (ii) (i) : e i 2 K[x] x α i f (x) (f(x), f (x)) K[x] 110. K 0 K[x] char(k) = 0 f (x) 0 f(x) deg(f(x)) > deg(f (x)) (f(x), f (x)) = K[x] 111. p x p t F p (t)[x] F p (t) d dx (xp t) = px p 1 = 0 ( 98 ) 110 0 8.2.. 112. L/K (1) α L K α p α,k (x) (2) L/K L K 113. L/K 2 (1) 2 L/K (2) 2 L = K(α) L/K α 2 K. (1) α L α K α K α K α p α,k (x) = x 2 + ax + b p α,k (x) = 2x + a 0 α K (2) α 2 K α L K 2 p α,k (x) = x 2 + α 2 = (x + α) 2 α ( 1 = 1 ) α 2 K α K p α,k (x) = x 2 +ax+b (a 0) p (x) = a 0 α p F p (t 1 p )/Fp (t) ( 98 111) ( ) 114. K 0 K 110

33 115. L/K (i) L/K (ii) K α 1,, α n L L = K(α 1,, α n ) (iii) K α L L = K(α) (iv) Hom K (L, K) = [L : K]. (i) (ii) : 19 (ii) (iii) 116. L/K K a L b L L = K(a, b) c L L = K(c) a, b K c K K K L K a b p a,k (x) = (x a 1 )(x a 2 ) (x a m ) p b,k (x) = (x b 1 )(x b 2 ) (x b n ) a i K (1 i m), a 1 = a b j K (1 j n), b 1 = b K d (b j b l )/(a i a k ) (i k) K K d c = b + da p b,k (c da) = 0 K(c) p b,k (c dx) p a,k (x) x a d p a,k (x) x a K(c) a K(c) b = c da K(c) K(a, b) K(c) K(c) K(a, b) K(a, b) = K(c) K L 8 L c L = K(c) K(c) = L = K(a)(b) a, b K 96 99 Hom K (K(c), K) = Hom K (K(a), K) Hom K(a) (K(a)(b), K) = [K(a) : K][K(a)(b) : K(a)] = [K(c) : K] 96 c K (iii) (iv) 96 (iv) (i) : α L 99 100 [L : K] = Hom K (L, K) = Hom K (K(α), K) Hom K(α) (L, K) [K(α) : K][L : K(α)] = [L : K] 100 Hom K (K(α), K) = [K(α) : K] 96 α K

34 117. L/K M (i) L/K (ii) L/M M/K (i) (ii) (ii) (i) : L/K 99 115 Hom K (L, K) = Hom K (M, K) Hom M (L, K) = [M : K][L : M] = [L : K] L/K L/K α L α M K N N(α)/N N/K N(α)/K α K 8.3.. 118. L/K L/K L K L 119. 2 2 120. Q(2 1 3 )/Q 2 1 3 Q 93 2 1 3 ω, 2 1 3 ω2 Q(2 1 3 ) 121. L/K (i) L/K (ii) K L α 1,, α n L L = K(α 1,, α n ) (iii) Aut(L/K) = Hom K (L, K). (i) (ii) : 19 (ii) (iii) : L K K α 1,, α n L 102 (iii) (i) : α L β K α K 95 L K ι ι(α) = β ι L β = ι(α) L 122. L/K M L/M 67

35 8.4.. 1 123. f(x) K[x] 0 K (1) K/K L f(x) f(x) L (2) f(x) K 124. f(x) K[x] 0 K f(x) K L L/K f(x) L/K f(x) = a(x α 1 ) (x α n ) L = K(α 1,, α n ) L f(x) K α i f(x) 121 L/K f(x) α i 115 125. Q f(x) = x 3 2 Q(2 1 3, ω) ω ω 2 + ω + 1 = 0 f(x) 2 1 3, 2 1 3 ω, 2 1 3 ω 2 ω = 2 1 3 ω/2 1 3 2 1 3 ω 126. ( ) Q 2 + 5 /Q 2 + 5 p 2+ (x) 5,Q ( ) Q 2 + 5, 1 2 + 5 = a+b 5 (a, b Q) a 2 +5b 2 = 2, 2ab = 1 4a 4 8a 2 +5 = 4 ( a 2 1 ) 2 + 1 = 0 Q ( 5 ) ( ) ( ) Q 2 + 5 [Q 2 + 5 : Q] = 4 2 + 5 Q 4 p 2+ 5,Q (x) = x4 4x 2 1 p 2+ (x) 5,Q p (x 2+ (x) = 2 + ) ( 5 x + 2 + ) ( 5 x 5,Q 2 ) ( 5 x + 2 ) 5 ( ) ( ) Q 2 + 5 Q 2 5 Q 2 + 5 ( ) ( ) Q 2 + 5 R Q 2 + 5 Q ( ) Q 2 + 5 2 + 5 2 5 = 1 p 2+ (x) 5,Q ( ) ( ) Q 2 + 5, 1 Q 2 + 5, 1 [ ( ) Q 2 + 5, 1 ] : Q = [ ( ) ] [ ( ) ( )] Q 2 + 5 : Q Q 2 + 5, 1 : Q 2 + 5 = 2 4 = 8

36 21. (1) n Q x n 2 (2) Q x 4 6x 2 2 22. (1) K p > 0 K K α K α K α K α (i) α K (ii) Hom K (K(α), K) = 1. (iii) e 0, α pe K. (2) K p > 0 L/K L/K L K (i) L/K (ii) Hom K (L, K) = 1. (iii) K α 1, α 2, L = K(α 1, α 2, ) (3) K p > 0 L/K [L : K] p (4) K p > 0 L/K L/K T L/K 23. K p > 0 L/K (1) L/K K i K i /K L/K i (2) L/K K s K s /K L/K s (3) K s K i = K L = K i K s ( ) (4) K i /K L/K i [L : K i ] = [K i : K] 24. K p > 0 L/K (i) L/K (ii) T L/K

37 9. 9.1.. 127. L/K L/K L/K L K Aut(L/K) Gal(L/K) 128. L/K M L/M 117, 122 129. L/K M K ML/M Gal(ML/M) Gal(L/K) L K K L ML/M σ Gal(ML/M) σ L = id L σ M σ = id ML [L : K] [M : K] Gal(ML/M) Gal(L/K) 130. L/K 2 (1) 2 L/K (2) 2 L = K(α) p α,k (x) = x 2 + ax + b 113 119 L/K a 0 Z/2Z 131. (1) 114 0 (2) 124 132. Q( 2, 3) Q Gal(Q( 2, 3)/Q) = Z/2Z Z/2Z Q( 2, 3) Q (x 2 2)(x 2 3) Q( 2, 3)/Q 4 ( 20) Q( 2, 3) Q( 3) σ σ( 2) = 2 Q( 2, 3) Q( 2) τ τ( 3) = 3

38 σ τ Q( 2, 3) Q σ 2 = τ 2 = id,, στ = τσ id Q( 2, 3) Q 1, 2, 3, 6 στ( 6) = σ(τ( 2 3)) = σ( 2τ( 3)) = σ( 2 3) = 2 3 = = τσ( 6) Gal(Q( 2, 3)/Q) 4 Z/4Z Z/2Z Z/2Z 2 2 Gal(Q( 2, 3)/Q) = Z/2Z Z/2Z Gal(Q( 2, 3)/Q) id, σ, τ, στ - 133. K L/K L N/K L K N L/K L/K ( 115) N ( 131 (2)) 134. 125 (1) Q(2 1 3 )/Q (2) Q(2 1 3, ω)/q Q(2 1 3 )/Q ω ω 2 + ω + 1 = 0 Gal(Q(2 1 3, ω)/q) 2 1 1 3 ω Q(2 3, ω) Q [Q(2 1 3, ω) : Q(ω)] = 3 2 1 1 3 Q(ω) 2 3, 2 1 3 ω, 2 1 3 ω 2 Q(ω) σ : Q(2 1 3, ω) Q(2 1 3, ω) σ(2 1 3 ) = 2 1 3 ω [Q(2 1 3, ω) : Q(2 1 3 )] = 2 ω Q ω, ω 2 Q(2 1 3 ) τ : Q(2 1 1 3, ω) Q(2 3, ω) τ(ω) = ω 2 σ 3 = id, τ 2 = id, τστ = σ 2 σ 3 (2 1 3 ) = σ 2 (2 1 3 ω) = σ(2 1 3 ω 2 ) = 2 1 3 ω 3 = 2 1 3 τ 2 (ω) = τ(ω 2 ) = ω 4 = ω τστ(2 1 3 ) = τσ(2 1 3 ) = τ(2 1 3 ω) = 2 1 3 ω 2 = σ(2 1 3 ) τστ(ω) = τσ(ω 2 ) = τ(ω 2 ) = ω 4 = ω = σ 2 (ω) G G =< σ, τ; σ 3 = τ 2 = id, τστ = σ 1 > G 6 G 3 S 3 σ τ G Gal(Q(2 1 3, ω)/q) Q(2 1 3, ω)/q 6 Gal(Q(2 1 3, ω)/q) = G = S3

39 135. L/K (i) L/K (ii) Aut(L/K) = [L : K]. 96 102 Aut(L/K) Hom K (L, K) [L : K] ( 121) ( 115) 9.2.. 136. (1) G L G L L (g, a) g(a) (i) (gh)(a) = g(h(a)). (ii) e(a) = a. e G (iii) g(a + b) = g(a) + g(b). (iv) g(ab) = g(a)g(b). (2) K L G K L G L (v) g(a) = a ( g G, a K) (3) G L (vi) g(a) = a ( a L) g = e 2 137. G L G Aut(L) K Aut(L) Aut(L/K) - 138. G L L G = {a L g(a) = a g G} K L G L/K L G G L

40 139. L G L K = L G L/K G G Aut(L/K) = Gal(L/K) 140. 139 α L (1) G α = {g G g(α) = α} G (2) g G g(α) G G α g(α) (3) f α (x) = g G/G α (x g(α)) K f α (α) = 0 g g (4) L f α (x) (1) (2) (3) (2) K G G/G α (4) g(α) L 139 : α L f α (x) 140 K f α (α) = 0 α K K α f α (x) L L/K [K(α) : K] deg(p α,k (x)) deg(f α (x)) G α L L/K K M [M : K] G L/K M/K 115 L/K L/K [L : K] G G Gal(L/K) [L : K] = Gal(L/K) G L/K G G Gal(L/K) 141. L/K K = L Gal(L/K) K L Gal(L/K) 139 [L : K] = Gal(L/K) = [L : L Gal(L/K) ]

142. k L = k(x 1,, x n ) n (n k[x 1,, x n ] ) s 1 = x 1 + + x n s 2 = i 1<i 2 x i1 x i2.. s n = x 1 x n n K = k(s 1,, s n ) L G = S n n G L G L L σ(f(x 1,, x n )) = f(x σ 1 (1),, x σ 1 (n)) L G = K L/K G s 1,, s n n G K L G G L 139 [L : L G ] = G = n!. [L : K] n!. n n 1 H = S n 1 n S n L H = k(x n )(t 1, t n 1 ) = K(x n ) t 1,, t n 1 x 1,, x n 1 K f(x) = x n s 1 x n 1 + s 2 x n 2 + ( 1) n s n f(x n ) = 0 [L H : K] n 41 [L : K] = [L H : K][L : L H ] n! 9.3.. 143. L/K (1) {H Gal(L/K) } {M M L/K } H L H Gal(L/M) M H 1, H 2 Gal(L/K) M 1, M 2 H 1 H 2 M 1 M 2 (2) H Gal(L/K) L H /K Gal(L/K)/Gal(L/L H ) Gal(L H /K)

42 (1) 139 141 (2) H L H /K ( 117)α L H α K g(α) (g G) H g(α) L H L H /K σ H H g 1 σg = τ H σ(g(α)) = g(τ(α)) = g(α) L H /K Gal(L/K) Gal(L H /K) H = Gal(L/L H ) H Gal(L/K) Gal(L H /K) Gal(L H /K) = Gal(L/K)/Gal(L/L H ) 144. 145. L/K M K 146. 4 Q ( 2, 3 ) /Q 132 Gal ( Q ( 2, 3 ) /Q ) = Z/2Z Z/2Z {e}, {e, σ}, {e, τ}, {e, στ}, Gal ( Q ( 2, 3 ) /Q ) 5 ( ) στ 6 = 6 Q ( 2, 3 ), Q ( 3 ), Q ( 2 ), Q ( 6 ), Q ( 2, 3 ) Q ( 2, 3 ) {e} Q ( 3 ) Q ( 6 ) Q ( 2 ) {e, σ} {e, στ} {e, τ} Q Gal ( Q ( 2, 3 ) /Q ) Q ( 2, 3 ) /Q Q 147. 6 Q(2 1 3, ω)/q ( 134) Gal(Q(2 1 3, ω)/q) = S 3 S 3 {e}, {e, τ}, {e, στ}, {e, σ 2 τ}, {e, σ, σ 2 }, S 3 Q(2 1 3, ω) {e} Q(ω) Q(2 1 3 ) Q(2 1 3 ω 2 ) Q(2 1 3 ω) {e, σ, σ 2 } {e, τ} {e, στ} {e, σ 2 τ} Q Gal(Q( 2, 3)/Q) {e, σ, σ 2 } Q Galois Q, Q(ω) Q(2 1 3, ω)

148. 8 Q 8 ( 2 + 5, 1 ) /Q ( 126) Z/8Z, Z/4Z Z/2Z, Z/2Z Z/2Z Z/2Z, H 4 ( H 4 = {±1, ±i, ±j, ±k; i 2 = j 2 = k 2 = 1, ij = ji = k}), D 4 (4 D 4 =< σ, τ; σ 4 = τ 2 = 1, τστ = σ 1 >) ( ( ) ) Gal Q 2 + 5, 1 /Q ( ) 2 + 5 Q Q ( ) 2 + 5, 1 /Q Gal /Q ( 126) ( 144) ( ( ) ) Q 2 + 5, 1 /Q H 4 ( D 4 ) Q 2 + 5 Q ( 5, 1 ) ( ( ) ) Q 4 Gal Q 2 + 5, 1 /Q 2 2 H 4 2 1 H 4 2 ( ( Gal Q 2 + 5, ) ) 1 /Q = D 4 ( ( ) ) Gal Q 2 + 5, 1 /Q ( Q 2 + 5, ) 1 = Q[x, y]/(x 4 4x p 1, y 2 + 1) ( ) 2 2 + 5, 1 Q 2 + 5, 1 ( ) Q 2 + 5, 1 σ τ { ( ) σ 2 + 5 = 2 5 σ ( 1 ) = 1 { ( ) τ 2 + 5 = 2 + 5 τ ( 1 ) = 1 σ τ 2 2 + 5 2 5 = 1 ( ) σ 2 5 = 2 + 5 ( ) τ 2 5 = 2 5 σ 43 2 + 5 1 σ σ σ σ 2 5 2 + 5 2 5 2 + 5 σ σ σ σ 1 1 1 1 σ 4 = τ 2 = e, σ 2 e τστσ σ τ σ τ 2 + 5 2 5 2 5 2 + 5 2 + 5 σ τ σ τ 1 1 1 1 1 1 τστ = σ 1 ( ) ( ( ) Q 2 + 5, 1 /Q Gal Q 2 + 5, 1 /Q) = D4

44 Q ( 5, 1 ) ( ) Q 2 + 5, 1 ( ) Q 2 + 5 ( ) Q 2 5 Q ( 5 ) Q ( 5 ) Q ( 1 ) Q ( ) Q 2 + 5 2 5 ( ) Q 2 + 5 + 2 5 {e} {e, σ 2 } {e, σ 2 τ} {e, τ} {e, στ}, {e, σ 3 τ} {e, σ, σ 2, σ 3 } {e, σ 2, τ, σ 2 τ} {e, στ, σ 2, σ 3 τ} D 4 Q ( 2 + 5, {e,στ} ( 1) = Q 2 + 5 2 ) 5 ( στ 2 + 5 2 ) ( 5 = σ 2 + 5 + 2 ) 5 = 2 5 + 2 + 5 ( Q 2 + 5 2 ) ( 5 Q 2 + 5, ) {e,στ} 1 1 = ( 2 + 5 + 2 5 ) 2 4 ( Q 2 + 5 2 ) ( 5 2 + ) ( 5 = Q 2 + 5, ) 1 ( ) 2 + 5 Q 2 + 5 2 5 2 ( x 2 2 + 5 2 ) 5 x 1 = 0 ( ) ( ) Q 2 + 5, 1 /Q 2 + 5 2 5 2 ( Q 2 + 5, {e,στ} ( 1) = Q 2 + 5 2 ) 5 2 D 4 {e}, {e, σ 2 }, {e, σ, σ 2, σ 3 }, {e, σ 2, τ, σ 2 τ}, {e, στ, σ 2, σ 3 τ}, D 4 Q 9.4.. R 2. (1) a 0 x 2 a R[x] 1 (2) R[x] R 1

45 (1) R (2) p ( p ) p = 2 A. G p G p H [G : H] p H p B. G p G G = G 0 G 1 G r 1 G r = {e} [G i 1 : G i ] = p (1 i r) 1. (1) C 2 1 2. (1) R 2 (2) C 2 (1) K/R 2 R 0 ( 43) K/R ( 114) K R L ( 133) G H G 2 K/R H G ( A) H L/R M M = R(α) ( 115) α R 1 ( 143) (2) R 2 (2) (1) : K C 2 K C K/C 2 K C B C 2 1 K = C 85 25. (1) Q Q(2 1 4 ) (2) Q Q( 3 + 11) 26. (1) Z/2Z Z/2Z Z/2Z Q (2) 4 H 4 Q

46 10. n µ n C 1 n Q Q(µ n ) n n 1 10.1.. 149. n ϕ ϕ(n) = { 1 if n = 1 (Z/nZ) if n > 1 150. n, m ϕ(mn) = ϕ(m)ϕ(n) 151. n, m 2 Z/mnZ Z/mZ Z/nZ a (a mod m, a mod n) (Z/mnZ) = (Z/mZ) (Z/nZ) 152. n = p e1 1 pe2 2 per r (r 1, p 1,, p r, e 1,, e r 1) r ϕ(n) = p ei 1 i (p i 1) i=1 150 153. p e ϕ(p e ) = p e 1 (p 1) (Z/p e Z) p 0 p e 1

47 10.2.. 154. n Q 1 n C Q(µ n ) n n n C 155. K n n (1) 2 n = K n = K 2n. (2) m n = K m K n. (1) µ 2n = µ n µ n (2) µ m µ n. 156. C 1 n µ n n µ n = {e 2πik n k Z/nZ} = Z/nZ 157. ζ C (i) ζ n n 1 (ii) k (Z/nZ), ζ = e 2πik n. (Z/nZ) Z/nZ 158. K C (i) K n (ii) K = Q(e 2πi n ). (iii) n ζ K = Q(ζ) - 159. n Φ n (x) = x ζ ζ:1 n C ϕ(n) Φ n (x) n

48 Φ n (x) 1 n n x n 1 = d n Φ d (x) n Φ n (x) 160. K n n K n /Q ζ C 1 n ρ : Gal(K n /Q) (Z/nZ) σ(ζ) = ζ ρ(σ) ρ ζ 1 : n x n 1 131 2 ζ : 157 n l (Z/nZ) ζ l σ(ζ l ) = σ(ζ) l = ζ lρ(σ) l ρ 1 n 3 ρ K n 4 ρ : f(x) ζ Q ( 78) f(x) n p f(ζ p ) = 0 f(ζ p ) = 0 104 ζ ζ p K n Z/nZ p ρ (Z/nZ) n ρ p n Φ n (x) = f(x)g(x) g(x) x n 1 F p [x] f(x) g(x) F p [x] f(x) g(x) F p F p Z[ζ] F p f(ζ p ) 0 ζ p 1 n g(ζ p ) = 0 ζ Z[ζ] F p ζ F p f(ζ) = 0, g(ζ) 0 g(ζ) p = g(ζ p ) = 0 F p g(ζ) = 0 f(ζ p ) = 0 ρ 161. n Φ n (x) [Q(ζ) : Q] = ϕ(n) = deg(φ n (x)) Φ n (x) Q

162. n K n = Q(ζ n ), ζ n = e 2πi n 2 ( 73) (1) K 2 = K 1 = Q 2 (2) K 6 = K 3 = Q( 3) 2 (3) K 4 = Q( 1) 2 (4) n = 5 Gal(K 5 /Q) = (Z/5Z) = Z/4Z 4 2 K 5 1 2 49 g = ζ 5 ζ 2 5 ζ 3 5 + ζ 4 5 K 5 g 2 = ζ5 2 ζ5 3 ζ 4 + 1 ζ5 3 + ζ5 4 + 1 ζ 5 ζ5 4 + 1 + ζ 5 ζ5 2 + 1 ζ 5 ζ5 2 + ζ5 3 = 4 ζ 5 ζ5 2 ζ5 3 ζ5 4 = 5 g = ± 5 K 5 2 Q( 5) 2 (5) n = 15 Gal(K 15 /Q) = (Z/15Z) = (Z/3Z) (Z/5Z) = Z/2Z Z/4Z 2 0 Z/4Z, Z/2Z 2Z/4Z, Z/2Z 2Z/4Z 155 K 3, K 5 K 15 K 15 2 Q( 3), Q( 5), Q( 15) n 163. n K n (i) n = 2 e p 1 p 2 p r. e 0 p 1,, p r 1 + 2 f (f 1) (ii) ϕ(n) 2 (iii) 2 Q = F 0 F 1 F s = K n (i) (iii) : 160 Gal(K n /Q) = Z/nZ Z/nZ 2 143 2 (iii) (ii) 160 (ii) (i) 152 (i) 2 3, 5, 17, 257, 27. m, n l K m K n = K l 28. 7 8 16 60 2 29. 2

50 11. 11.1.. 2 ( ) ( 1) 164. Ω 2 (1) Γ = Ω (a), (b), (c) Γ Γ Ω (a) Γ 2 2 l, l l l Γ (b) Γ 2 l Γ 2 π l π Γ (c) Γ 2 2 π, π π π Γ (2) Ω 2 2 (3) Ω 2 (4) Ω 3 A, B, C ABC Ω 2 2 2 B B A A A, B Ω A, B Ω 11.2.. 3 2 165. O, P, A, B O P {O, P, A, B} OP Q OQ = AB A = B A B AB OP A OP A OA (b) B OA O A B OP

51 B B B A A A B O P O P (b) O AB B A B A B O OA 2 M O A OA 2 D E (c) M DE OA (a) D Q B O M A B O M A E M MB OP B Q OQ = AB (b) Q 166. l l P P l 167. 2 2

52 168. 1 α, β α ± β, αβ, α/β (β 0) αβ S R α β O A P Q αβ AR//QS OA = 1, OP = α, OQ = β QS = αβ 169. 1 170. 1 α α P α B α M O 1 A OA = 1, OB = α M AB OP AB OP = α 11.3.. Ω 0 1 Ω = {0, 1}

53 171. α C (i) α (ii) α (iii) α α 172. (1) α, β C α ± β, αβ, α/β (β 0) (2) a, b, c C, (a 0) 2 ax 2 + bx + c = 0 (1) (2) (1) x 2 = c 173. Ω 0, 1 C α C (i) α Ω (ii) α K n Q(Ω) = K 0 K 1 K 2 K n 1 K n [K j : K j 1 ] = 2 (1 j n) (ii) (i) 172 (i) (ii) : (a) (b) (c) 3 2 (a) 2 (b) 0 r a ± r 2 a 2 (c) 0 v > 0 r s r 2 x 2 = s 2 (v x) 2 1 (b) 2 174. α C [Q(α) : Q] 2 11.4. n. n 1 1 n n 1 n 175. n 3 (i) n (ii) n = 2 e p 1 p 2 p r. e 0 p 1,, p r 1 + 2 f (f 1)

54 163 176. 5 = 1 + 2 2 5 ζ 5 = e 2πi 5 x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) f(x) = x 4 + x 3 + x 2 + x 1 + 1 f(ζ 5 ) = 0 t = x + x 1 g(t) = x 2 f(x) = x 2 + x + 1 + x 1 + x 2 = t 2 + t 1 g(t) = 0 t = 1± 5 2 2 Q Q( 1 + 5 ) Q(ζ 5 ) 2 5 2Re(ζ 5 ) = ζ 5 + ζ 5 = 1+ 5 2 = 2a 2Re(ζ 2 5) = ζ 2 5 + ζ 2 5 = 1 5 2 = 2b a b 5 + 1 5 + 1 5 5 1 5 1 1 0 1 2 5 b a 0 1 4 i ζ 5 i ζ 5 ζ 2 5 ζ 2 5 b 0 a 1 0 1 ζ 2 5 ζ 2 5 ζ 5 ζ 5 11.5. 3. 3 (a) : 2 (b) 3 : 3 (c) :

55 (a) x 3 = 2 x = 2 1 3 [Q(2 1 3 ) : Q] = 3 (b) e 2πi 3 1 3 2π 3 3 2π 9 2π 9 9 (c) x 2 = π π π π 30. 17

56 12. 19 4 5 5 12.1.. d d 177. K f(x) K[x] d (d 1) L f(x) K f(x) Gal(L/K) Gal(L/K) d S d f(x) K Gal(L/K) Gal(L/K) f(x) 104 178. K f(x) K[x] d (d 1) K K f(x) = (x α 1 )(x α 2 ) (x α d ) f(x) f D f f = (α i α j ) i<j D f = i<j d = 1 D f = f = 1 (α i α j ) 2 = 2 f D f = 0 f(x) 179. (1) f(x) = x 2 + ax + b D f = a 2 4b (2) f(x) = x 3 + ax + b ( 3 ) D f = (4a 3 + 27b 2 ) 180. L f(x) K Gal(L/K) S d (1) f L, D f K. (2) f K Gal(L/K) A d. A d d D f α 1,, α d f α 1,, α d

57 181. f(x) = x 3 + ax + b K[x] L f(x) K (1) D f (K ) 2 f K Gal(L/K) = S 3. (2) D f (K ) 2 f K Gal(L/K) = A 3. f(x) [L : K] 3 A 3 182. (1) f(x) = x 3 3x + 3 Q[x] p = 3 D f = (4 ( 3) 3 + 27 3 2 ) = 135 (Q ) 2 f(x) L Gal(L/Q) = S 3 (2) f(x) = x 3 3x + 1 Q[x] 3 F 3 [x] D f = (4 ( 3) 3 + 27 1 2 ) = 81 = 9 2 (Q ) 2 f(x) L Gal(L/Q) = A 3 = Z/3Z 183. L ( C) Q f(x) = x 5 20x + 5 Gal(L/Q) = S 5 p = 5 f(x) Gal(L/Q) 5 Gal(L/Q) 5 f(x) 3 2 C f(x) L Gal(L/Q) S 5 5 Gal(L/Q) = S 5 184. p p p p p (123 p) (12) (123 p) i (12)(123 p) i = (i + 1, i + 2)

58 12.2.. ( ) 185. K k K (char(k) = 0 k = Q char(k) = p > 0 k = F p ) (1) L/K K = M 0 M 1 M r 1 M r = L 1 n j M j = M j 1 (aj ) (a j M j 1 ) 1 n j a j x nj a j (2) K x n + c 1 x n 1 + + c n = 0 ( ) k(c 1,, c n ) L k(c 1,, c n ) x n + c 1 x n 1 + + c n L 186. M/K L/M L/K 187. (1) 2 2 x 2 + ax + b = 0 (2) Q x 3 = 2 (1) x 2 + ax + b = (x + a 2 )2 a2 4b 4 x = a± a 2 4b 2 (2) x 3 2 Q( 3, 2 1 3 ) Q 12.3.. 188. K 1 n ζ L/K n (i) L/K (ii) a K, L = K(a 1 n ). (i) (ii) : σ Gal(L/K) 18 t L θ θ = t + ζσ(t) + ζ 2 σ 2 (t) + + ζ n 1 σ n 1 (t) 0 σ(θ) = ζ 1 θ θ n = ±θσ(θ)σ 2 (θ) σ n 1 (θ) K f(x) = x n θ n K L = K(θ) (ii) (i) σ(a 1 n ) = ζ 1 n Gal(L/K) σ 189. K 0 ζ m 1 m K n = K(ζ 1, ζ 2, ζ 3, ζ n ) K

59 186 K m /K m 1 Gal(K m /K m 1 ) Gal(Q(ζ m )/Q) ( 129) K m /K m 1 h m ζ h K m 1 188 190. Q x n = 1 12.4.. 191. G G = D 0 D 1 D r = {e} D j = [D j 1, D j 1 ] 192. G (i) G (ii) G (iii) G G = G 0 G 1 G r = {e} G j G j 1 G j 1 /G j G = G 0 G 1 G r = {e} G j G j 1 G j 1 /G j 193. 194. (1) (2) p p (9.4 B) p (3) 3 4 S 3, S 4 A n n S 3 A 3 {e} S 4 A 4 {(12)(34), (13)(24), (14)(23), e} {e}

60 195. G G (G {e} ) 196. 197. (1) [S n, S n ] = A n. (2) n 5 A n 198. n 5 S n A n 12.5.. 199. K 0 L/K (i) L K (ii) Gal(L/K) (i) (ii) : K 1 n M = K(a 1 1,, a 1 1 1 nr r ) ( x nj n a j K(a 1 1,, a n j 1 j 1 ) ) L M K M M/K Gal(M/K) Gal(L/K) M/K r r = 1 M = K(ζ n, a 1 n ) ζ n 1 n M/K(ζ n ) K(ζ n )/K Gal(M/K) r > 1 M 1 1 n K(a 1 1,, a n r 1 r 1 ) M M M (ζ nr, a 1 nr ) r = 1 Gal(M (ζ nr, a 1 nr )/M ) Gal(M/M ) Gal(M /K) Gal(M/K) (ii) (i) : 189 186 K 1 [L : K]! 188 L/K

61 200. 0 x n + c 1 x n 1 + + c n 1 x + c n = 0 Q(c 1,, c n ) x n + c 1 x n 1 + + c n 1 x + c n L Gal(L/Q(c 1,, c n )) 201. (1) n 4 n x n + c 1 x n 1 + + c n = 0 (2) n 5 n x n + c 1 x n 1 + + c n = 0 177 Q(c 1,, c n ) S n (1) 4 S 4 (2) x n + c 1 x n 1 + + c n = 0 Q(c 1,, c n ) n 142 Q(c 1,, c n ) n S n C c 1,, c n 202. x 3 3x + 1 = 0 C x 3 3x + 1 = 0 3 α, β, γ (α < β < γ) 182 (2) Q(α)/Q A 3 = Z/3Z ζ = 1+ 3i 2 1 3 [Q(α, ζ) : Q] = 6 x 3 3x + 1 Q(ζ) Gal(Q(α, ζ)/q(ζ)) = Gal(Q(α)/Q) σ σ(α) = β D = 81 188 [Q(α, ζ) : Q] = 6 = (α β)(α γ)(β γ) = 9 θ = α + ζβ + ζ 2 γ θ = α γ + ζ(β γ) 0 θ θ 3 = α 3 + β 3 + γ 3 + 3ζ(α 2 β + β 2 γ + γ 2 α) + 3ζ(αβ 2 + βγ 2 + γα 2 ) + 6αβγ = 9 + 3ζ +3 2 3ζ 2 3 2 = 27+27 3i 2 Q(ζ) θ = 3 ( 1 ) 1 3 3i 2 α Q( 3) θ ( π 3π 2 α = l + mθ + nθ 2 ) θ = α + ζβ + ζ 2 γ m = 1 3 α2 + αβ + β 2 = αβ + (α + β) 2 = αβ βγ γα = 3 l = 0, n = 3 θ 3 α = 1 3 θ + 1 + 3i θ 2 Q( 3i, θ) = Q(θ) 18

62 n 5 n 183 x 5 20x + 5 = 0 S 5 S n (n 5) 31. Q 3 S 3 32. f(x) = x 4 + ax 2 + bx + c K[x] 4 D f f f(x) G f(x) L K Gal(L/K) (1) S 4 G (2) D f (3) D f (K ) 2 G S 4 (4) D f (K ) 2 G S 4 (5) b = 0 G S 4 (6) b 0 f(x) = (x 2 ux + s)(x 2 + ux + t) K s + t K G S 3 A 3 33. f(x) = x 4 4x + 2 Q[x] f(x) Q 34. Q 4 Q A 4 35. Q x 3 3x + 3 = 0