S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.

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1 () (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2 + 3) 1.3. L K i (i I) i I K i L 1.2. K L L S K S L K(S) K S S = {s 1, s 2,..., s n } K(S) K(s 1, s 2,..., s n ) α L K(α) K 1.4. n K(S) (S = {s 1, s 2,..., s n } L) { } f(s1,..., s n ) K(S) = g(s 1,..., s n ) f, g K[X 1,..., X n ], g(s 1,..., s n ) 0 1

2 S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1. K M L (1) {α 1, α 2,..., α m } M K {β 1, β 2,..., β n } L M {α i β j 1 i m, 1 j n} L K (2) () L/M M/K L/K [L : K] = [L : M][M : K]. (1) () {β 1, β 2,..., β n } M L x y j M(1 j n) x = y j β j 1 j n y j {α 1, α 2,..., α m } M K z ij K(1 i m) y j = z ij α i 1 i m {α i β j 1 i m, 1 j n} K L () c ij K c ij α i β j = 0 1 i m 1 j n

3 () 3 1 j n 1 i m c ij α i β j = 0 i c ijα i M {β j } n j=1 M j c ij α i = 0. 1 i m {α i } m i=1 K c ij = 0 (2) (1) 1.6. K L K L K, L 1.7. M 1, M 2 K L [M 1 : K] [M 2 : K] M 1 M 2 = K K L α L I α = {f(x) K[X] f(α) = 0} K[X] (i) I α = (0) α K (ii) I α (0) α K I α 1 α K Irr(α, K, X) 1.8. K α K α α K 1.9. Q Q Q (1) 2 (2) ω (1 3 ) ω 1 3 (1) ω Q( 3 2) (2) [Q( 3 2, ω) : Q] (3) Q( 3 2, ω) Q α = 3 1 (1) α (2) α L/K L K L/K L/K

4 4 Q e π C/R (1) α K K(α)/K (2) α, β K α ± β, αβ, αβ 1 (β 0) K (3) L K K Q 1.2. L K L K. ( ) [L : K] [L : K] = 1 [L : K] 2 K L α [L : K(α)] < [L : K] L K(α) β 1, β 2,..., β n L = K(α, β 1, β 2,..., β n ) α β i (1 i n) K ( ) n n = 1 L = K(α 1,..., α n 1, α n ) [K(α 1,..., α n 1 ) : K] < L/K(α 1,..., α n 1 ) Irr(α n, K, X) 1.1 [L : K] (1) K, K f : K K f f ( ) K K K = K (2) L/K, L /K ϕ : L L f = ϕ K K K ϕ f K = K id K K ϕ ϕ(x) = x (x K) ϕ K L = L ϕ K f : K K (1) K x f( x) = f(x) (2) K x 0 f(x 1 ) = f(x) 1 (3) Q K ( K 0) f Q (1) L K α L K f(x) L K ϕ : K(α) L ϕ(α) f(x) (2) Q ϕ : Q(α) C (3) ϕ( 3 2) = 3 2 ω L = Q( 3 2) (4) L = Q( 3 2) Q

5 () K f(x) K L K L[X] K[X] f(x) = γ(x α 1 )(X α 2 )... (X α n ); α 1, α 2,..., α n, γ L, γ 0 L f(x) 2.1. f(x) K[X] f(x) L K. f(x) n n = 1 K n > 1 f(x) = f 1 (X)f 2 (X) f 1 (X), f 2 (X) K[X] f 1 (X) L 1 K f 2 (X) L 2 L 1 L 2 K f(x) f(x) K 1 = K[X]/(f(X)) K X K 1 f(x) K 1 [X] f(x) f(x) 1 n 1 f(x) 2.2. L K f(x) K[X] α 1, α 2,..., α n L f(x) L f(x) K(α 1, α 2,..., α n ) f(x) K f(x) K[X] 2.2. σ K 1 K 2 f(x) K 1 L 1 K 1 f(x) L 2 K 2 σf(x) σ L 1 L 2. n = [L 1 : K 1 ] n = 1 σ n > 1 K 1 [X] f(x) 2 g(x) K 2 [X] σg(x) σf(x) g(x) α σg(x) β σ : K 1 (α) K 1 [X]/(g(X)) K 2 [X]/(σg(X)) K 2 (β) σ L 1 K 1 (α) f(x) L 2 K 2 (β) σf(x) L 1 L 2 σ σ 2.1. (1) Q X (2) Q X 3 2 L [L : Q] (3) L = Q( i) Q X 4 + 2X

6 K α, β K K 1. a + b i (a, b R) a b i R Q 2.3. L f(x) K[X] (1) f(x) α β K (2) σ(α) = β K σ : L L. (2) (1). g(x) = Irr(α, K, X) g(x) K g(β) = g(σ(α)) = σ(g(α)) = 0 β g(x) (1) (2). K σ : K(α) = K(β) L K K(α) f(x) K(α) = L K(β) L [K(β) : K] = [L : K] K(β) = L f(x) K(α) 2 f 1 (X) K(α) f 1 (X) α 1 K(β) f σ 1 (X) β 1 σ K K(α, α 1 ) = K(α)[X]/(f 1 (X)) = K(β)[X]/(f σ 1 (X)) = K(β, β 1 ) K(α, α 1,... ) L 2.3. f(x) K α 1, α 2,..., α t f(x) L f(x) K σ : L L α 1, α 2,..., α t 2.4. Q X 3 2 L σ : L L Q (1) L = Q( 3 2, ω) (2) σ( 3 2) σ(ω) (3) τ( 3 2) = 3 2 ω, τ(ω) = ω Q τ : L L τ X L/K (1) f(x) K[X] L L (2) α L α K L (3) α L α K L L/K

7 () (1) Q( 2)/Q (2) Q( 3 2)/Q 3.2. (1) L/K L, K M L/M (2) M, N L/K M/K N/K MN/K (3) M, N L/K M/K MN/N 3.2. L/K (1) L/K (2) L M K σ : M M σ(l) L. (1) (2). L = K(α 1, α 2,..., α n ) α i K f i (X) f i (σ(α i )) = σ(f i (α i )) = 0 σ(α i ) α i K σ(α i ) L (2) (1). α L f(x) α K M f(x) 2.3 α K β σ(α) = β M K β = σ(α) σ(l) L β L 3.3. L/K L K. ( ) 1.4 L K α i (i = 1, 2,..., n) α i f i (X) f(x) = f 1 (X)f 2 (X) f n (X) L/K L f(x) K ( ) L f(x) K f(x) α 1, α 2,..., α n L = K(α 1, α 2,..., α n ) L K σ f(σ(α i )) = σ(f(α i )) = 0 σ(α i ) f(x) σ(l) L 3.2 L/K 3.3. Q( 4 2, i) Q( 4 2) Q (1) Q( 4 2, i)/q (2) Q( 4 2, i)/q( 4 2) (3) Q( 4 2)/Q 3.4. Q( 4 2)/Q( 2), Q( 2)/Q Q( 4 2)/Q

8 (1) K α Irr(α, K, X) K (2) L/K L K f(x) = a n X n + a n 1 X n a 1 X + a 0 K[X] f (X) f (X) = na n X n 1 + (n 1)a n 1 X n a 1 f(x), g(x) K[X] a, b K (4.1) (4.2) (af(x) + bg(x)) = af (X) + bg (X) (f(x)g(x)) = f (X)g(X) + f(x)g (X) 4.1. (1) (4.1) (2) (4.2) ( f(x), g(x) ) 4.2. K f(x), g(x) (1) (g(x) k ) = kg(x) k 1 g (X) (0 k Z) (2) (f(g(x))) = f (g(x))g (X) f(x) g(x) g(α) 0 f(x) = (X α) e g(x), f(x) α e 2 α e = 1 f (X) = g(x) + (X α)g (X) f (α) = g(α) 0 e 2 f (X) = e(x α) e 1 g(x) + (X α) e g (X) f (α) = 0 α f(x) f(α) = f (α) = K 0 L/K. α L f(x) f(x) 1 K 0 f (X) 0 f (X) f(x) f(x) f (X) f(x) (4.3) a(x)f(x) + b(x)f (X) = 1 a(x), b(x) K[X] K[X] f(α) = f (α) = 0 (4.3)

9 () K L L = L \ {0} 1 L K K L K L K 2 L = K(α, β) α, β K f(x), g(x) α, β K α 1 = α, α 2,..., α m f(x) β 1 = β, β 2,..., β n g(x) f 1 (X) = f((α 1 cβ 1 ) + cx) g(x) β 1 0 c K f 1 (X) β 1 β 1 + α i α 1 c (i = 2, 3,..., m) (i, j) : 2 i m, 2 j n β 1 + α i α 1 c β j c 0 K c θ = α cβ(= α 1 cβ 1 ) h(x) β K(θ) h(x) f 1 (X) g(x) f 1 (X) g(x) β h(x) = X β h(x) K(θ) β K(θ) α = θ + cβ K(θ) K(α, β) K(θ) K(θ) K(α, β) 5. L Aut(L) L 5.1. Aut(L) 5.1. (1) L Aut(L) G F(G) = {a L σ(a) = a σ G} L G (2) L/K L K G(L/K) = {σ Aut(L) σ(a) = a a K} 5.2. (1) F(G) L (2) G(L/K) Aut(L) 1

10 10 Aut(L) G 1, G 2 L K 1, K 2 G 1 G 2 F(G 1 ) F(G 2 ) K 1 K 2 G(L/K 1 ) G(L/K 2 ) 5.3. (1) σ : C C; a + b i a b i (a, b R) R G(C/R) = {id, σ} (2) G(Q( 2)/Q) 5.1. (1) L Aut(L) G (a) (b) (2) L/K (c) G(L/F(G)) G F(G(L/F(G))) = F(G) F(G(L/K)) K (d) G(L/F(G(L/K))) = G(L/K). (a) (c) (b) (a) F F(G(L/F(G))) F(G) (c) K F(G) F(G(L/F(G))) F(G) (d) (b) 5.4. (1) G(Q( 3 2)/Q) = {id} (2) F(G(Q( 3 2)/Q)) Q L/K G(L/K) L/K 6.1. L M K L/K L/M 6.1. L/K G (1) L (2) [L : K] = G (3) F(G) = K.(1) 4.2 L = K(α) α L L/K f(x) = Irr(α, K, X) L L f(x) K L/K f(x) (2) L = K(α) L K σ σ(α) 2.3

11 () 11 α K β σ(α) = β K σ G α K f(x) (3) L F(G) K L/F(G) 5.1(d) G(L/F(G)) = G (2) K F(G) [L : F(G)] = G [L : K] = [L : F(G)][F(G) : K] [F(G) : K] = 1 2 K f(x) L f(x) α 1, α 2,..., α n L/K G ( ) α1... α σ i... α n σ(α 1 )... σ(α i )... σ(α n ) α 1, α 2,..., α n 6.2. Q( d)/q d 6.3. L = Q( 2, i) (1) L Q (2) σ L Q σ( 2) = 2, σ(i) = i τ L Q τ( 2) = 2, τ(i) = i σ τ G = σ, τ G (3) G(L/Q) = G (4) F(G) = Q 6.2. L/K {[K(α) : K] α L} m [K(µ) : K] = m µ L L = K(µ). α L L 4.2 K(µ, α) = K(ν) ν L K(µ) K(ν) [K(µ) : K] [K(µ) : K] = [K(ν) : K] K(µ) = K(ν) α K(µ) = L 6.3. L G Aut(L) K = F(G) (1) L/K (2) [L : K] = G (3) G(L/K) = G. (1) α L {α 1, α 2,..., α r } = {σ(α) σ G} ( ) f(x) = (X α 1 )(X α 2 )... (X α r ) σ G σ(f) = f f(x) K[X] Irr(α, K, X) f(x) L L/K 2 (X a) n

12 12 {[K(α) : K] α L} G = deg f(x) 6.2 L/K (6.1) [L : K] G (2) K (6.2) G G(L/K) (1) 6.1(2) G G(L/K) = [L : K] (6.1) G = G(L/K) = [L : K] (3) (6.2) ((2) ) 6.4. Q X 3 2 L (1) L = Q( 3 2, ω) ω = ( i)/2 (2) σ G(L/Q) { 3 2, 3 2 ω, 3 2 ω 2 } (3) L/Q G(L/Q) 3 S 3 (4) σ L Q σ( 3 2) = 3 2 ω, σ(ω) = ω τ L Q τ( 3 2) = 3 2, τ(ω) = ω 2 σ, τ (5) H = {ρ G(L/Q) ρ( 3 2) = 3 2} K = F(H) (6) L/K (7) K/Q 6.5. Q( i)/q (1) α = i Q α f(x) = Irr(α, Q, X) (2) f(x) L = Q( i) (3) f(x) α 1 = α, α 2, α 3, α 4 (4) ( ). L/K G (1) L/K M G H M G(L/M) F(H) H (2) M H L/M H [L : M] = H, [M : K] = G : H (3) M H M/K H G G(M/K) = G/H.. (1),(2)

13 () 13 (3) ( ) M/K u M σ G σ(u) M τ H τ(σ(u)) = σ(u) σ 1 τσ G(L/M) = H H G ( ) τ H, σ G σ 1 τσ H τσ(u) = σ(u) σ(u) M M/K M/K M/K G(L/K) G(M/K); σ σ M x 3 + ax 2 + bx + c = 0 x = y a/3 (8.1) y 3 + 3py + q = p = b/3 a 2 /9, q = 2a 3 /27 ab/3 + c y = u + v u 3 + v 3 + 3(u + v)(uv + p) + q = 0 (8.2) u 3 + v 3 = q (8.3) uv = p u, v y = u + v (8.1) (8.3) 3 (8.4) u 3 v 3 = p 3 (8.2),(8.4) u 3, v 3 x 2 + qx p 3 = 0 u 3, v 3 = q ± q 2 + 4p 3. 2 y = u + v + u 3 (8.3) (u, v) 3 q + q 2 + 4p 3, 3 q q 2 + 4p 3, q + q 2 + 4p 3 ω, 3 q q 2 + 4p 3 ω 2, q + q 2 + 4p 3 ω 2, 3 q q 2 + 4p 3 ω 2 2

14 14 ω 1 y = u + v x = y a/ (1) X 3 + 3X 1 = 0 (2) X 3 3X 1 = 0 (3) X 3 + 6X 7 = A, B A = , B = (1) X 3 + X 2 = 0 (2) A + B X 3 + X 2 = 0 A + B (3) X 3 + X 2 = X 3 +ax 2 +bx+c = 0 a, b, c f(x) = X 3 + ax 2 + bx + c α 1, α 2, α 3 a, b, c K = Q(a, b, c) f(x) L L = K(α 1, α 2, α 3 ) = Q(a, b, c, α 1, α 2, α 3 ) = Q(α 1, α 2, α 3 ) Aut(L) G {α 1, α 2, α 3 } {1, 2, 3} G F(G) = K K(α 1, α 2, α 3 )/K G = S 3 F(G) = K α 1, α 2, α 3 α 1 + α 2 + α 3 = a, α 1 α 2 + α 2 α 3 + α 3 α 1 = b, α 1 α 2 α 3 = c A 3 = {id, (1, 2, 3), (1, 3, 2)} F(A 3 ) = (α 1 α 2 )(α 1 α 3 )(α 2 α 3 ) D = 2 f(x) = 0 D = 27(4p 3 + q 2 ) K F(A 3 ) = K( ) K( ) K M = K( ) S 3 > A 3 > {id} K M L L M 3 A 3 ω K ω p, q, u, v p, q, ω K, uv = p = (1 ω) 3 (u 3 v 3 ) = (1 ω) 3 3 q 2 + 4p 3 M = K( ) = K( 3 q 2 + 4p 3 ), L = M(u) = M(v)

15 () f(x) = X 3 + 3pX + q Q L Q f(x) { A 3 D 2 G(L/Q) = D 2 S 3 D = 27(4p 3 + q 2 ) f(x) = Q f(x) = X 3 3X + 1 L f(x) L/Q σ S n f(x 1, X 2,..., X n ) f(x σ(1), X σ(2),..., X σ(n) ) K(X 1,..., X n ) S n σ S n σf = f f K(X 1,..., X n ) n () n (X + X 1 )(X + X 2 )... (X + X n ) n k T k T k = ( ) X i U i U U k {1, 2,..., n} T k X 1, X 2,..., X n k (k = 1, 2,..., n) 2. a, b, c a, b, c, d a + b + c, ab + ac + bc, abc a + b + c + d, ab + ac + ad + bc + bd + cd, abc + abd + acd + bcd, abcd 8.2. T 1, T 2,..., T n X 1, X 2,..., X n L = K(X 1, X 2,..., X n ) M = K(T 1, T 2,..., T n ) L/M S n. S n L K F F(S n ) = F L/F S n L F M F = M T k X 1, X 2,..., X n k f(x) = X n T 1 X n ( 1) n T n f(x) = (X X 1 )(X X 2 )... (X X n )

16 16 L f(x) M [L : M] n! L/F [L : F ] = S n = n! ( 1.1) [L : M] 1 [F : M] = [L : F ] 1 F = M 8.4. (1) f/g f g (2) ( ) G G = G 0 G 1 G 2... G i G i 1 (i = 1, 2, 3,... ) G G = G 0 G 1 G 2 G n = {e} G i 1 /G i G 3. G H K K G 4 S 4 ρ = ( ) σ = (1 2)(3 4) G G = {id, ρ, ρ 2, ρ 3, σ, σρ, σρ 2, σρ 3 } ( 4 ) ρ 4 = σ 2 = id ρσ = σρ 1 G G A 4 H H = {id, ρ 2, σ, σρ 2 } K = {id, σ} G H K K G 8.5. G H G K G K H H 8.3. K 0 L f(x) K[X] f(x) = 0 L/K L f(x) F [X] G(L/F ) n S n

17 () p p f(x) Q 2 p 2 f(x) Q L G = G(L/Q) p S p. G G Q f(x) Q p p [L : Q] = G G p p p p S p p G = S p ( ) 5. f(x) = X 5 5X + 1 Q[X] Q S 5 f(x) = (1) n f(x) = X n + a n 1 X n a 0 K = Q(a n 1,..., a 0 ) K f(x) L L/K n S n (2) A n 8.7. A 5 a a n = 1 n a ord(a) a n = 1 n a 8.6. a, b (1) a n = 1 ord(a) n. (2) ord(a) = mn ord(a m ) = n. (3) a, b m = ord(a) n = ord(b) ord(ab) = mn.. (1) ( ). m = ord(a) q, r n = qm + r, 0 r < m 1 = a n = a qm+r = (a m ) q a r = a r m r = 0 ( ). m = ord(a) n = mq q a n = (a m ) q = 1. (2) p = ord(a m ) (a m ) n = a mn = 1 (1) p n a mp = (a m ) p = 1 (1) mn = ord(a) mp n p (3). p = ord(ab) m, n (ab) mn = (a m ) n (b n ) m = 1

18 18 (1) p mn mn p b np = (b n ) p = 1 a np = a np b np = (ab) np = ((ab) p ) n = 1 (1) m np m n m p n p p m, n mn 8.7. G. G n n = 1 n > 1 n n n = p e 1 1 pe pes s p 1, p 2,..., p s i p ei i n G 8.6(2) p e i i a i 8.6(3) ord(a 1 a 2... a s ) = n G n i (1 i s) a G p e pe i 1 i 1 pe i 1 i p e i+1 i+1... pe s s = n/p i a n/p i = 1 X n/p i = 1 n/p i G a, b a 1 b 1 ab [a, b] a, b G H, K [h, k](h H, k K) [h, k] h H, k K [H, K] [G, G] G [a, b] (a, b G) G g g 1 [a, b]g = [g 1 ag, g 1 bg] [a i, b i ] (a i, b i G, 1 i n) g 1 [a 1, b 1 ][a 2, b 2 ]... [a n, b n ]g = g 1 [a 1, b 1 ]gg 1 [a 2, b 2 ]g... g 1 [a n, b n ]g = [g 1 a 1 g, g 1 b 1 g][g 1 a 2 g, g 1 b 2 g]... [g 1 a n g, g 1 b n g] [G, G] G G H (1) G/H (2) bah = abh (a, b G) (3) a 1 b 1 ab H (a, b G) (4) [G, G] H G [G, G] G/H H

19 () 19 G D 0 (G) = G G D i (G) = [D i 1 (G), D i 1 (G)] (i > 0) D 0 (G) D 1 (G) D 2 (G) G G. G = G 0 G 1 G 2 G n = {e} G i 1 /G i 8.8 G i D i (G) D n (G) = {e} A n (n 3) 3. (i j), (k l) 0 3 ( 0 1 ) {i, j} {k, l} {i, j} {k, l} 1 () j = l (i j)(k j) = (i j k) 3 3 (i j)(k j) = (i k j)(i k l) 6. i, j, k, l, m (m l k) 1 (i j k) 1 (m l k)(i j k) = (j k l) n 5 j, k, l i, m 3 3 D(A n ) = A n A n p S p τ p σ. S n (1 2) ( n) τ (1 2) σ 1 k 2 σ k σ k 1 p σ k 1 = ( ) 3,..., p τ σ k 1 S n τ σ S p [1] ( )1983 [2] ( )1989 [3] John M. Howie Fields and Galois Theory (Springer)2006

16 B

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..