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2 1 ( ),,. R R R R : (a, b) a + b, R R R : (a, b) a b, R. (R1) R +. (R2) R a (b c) = (a b) c. (R3) R a (b + c) = a b + a c, (a + b) c = a c + b c. (R4) R 0, R e 0, a R e a = a e = a. e R, 1 1 R. 0 0 R. R a b = b a, R.. R, R Z R Z R = {a R b R a b = b a }. Z R R. Z R R. R. R a, a b = b a = 1 R b R, a. b a 1, a. R R. R. R. K K. (K1) K. (K2) K = K {0}, K. 2

3 K K. K, K. K. A K, λ K, a, b A (λ a) b = a (λ b) = λ (a b), A K K. A K, K A : λ λ1 A. K 1 A K, K A. R, M +. ( ) M R M : (u, a) ua (1), a, b R, u, v M, M R. (M1) (u + v)a = ua + va (M2) u(a + b) = ua + ub (M3) u(a b) = (ua)b (M4) u1 R = u R. R 1 R 2, M R 1 R 2., a 1 R 1, a 2 R 2, u M (a 1 u)a 2 = a 1 (ua 2 ), M (R 1, R 2 ). R. R M, {0} M R, R ( )R, R R ( ).. R, {0} R,. 3

4 2 2.1 R. M R. End R (M) End R (M) = {f : M M f R } (f + g)(u) = f(u) + g(u), (f g)(u) = f(g(u)) (u M)., End R (M). M. End R (M) M M : (f, u) f(u), M End R (M). R, M (End R (M), R). End R (M) Aut R (M), M. Aut R (M) = {f End R (M) f }. 1 (Schur) M R, End R (M). f End R (M), f 0. f 0, Kerf M {0} Imf. M, Kerf = {0} Imf = M, f, f Aut R (M). 2.2 R, m, n, M m,n (R) R m n. m = n, M n,n (R) M n (R). M n (R), R n. i = 1, 2,, n, j = 1, 2,, n, e ij (i, j) 1, 0 M n (R). e ij M n (R). e ij. e ij e kl = { eil (j = k) 0 Mn (R) (j k) (1 i, j, k, l n). 4

5 e ij M n (R)e ij e ij M n (R) 0 0 a 1i 0 0 M n (R)e ij =..... j a 1i,, a ni R, 0 0 a ni e ij M n (R) = a j1 a jn i a j1,, a jn R M n (R) R. R a 1 R n a 2 =. a 1, a 2,, a n R a n. R n 1 0 e 1 = 0, e 2 = ,, e n = e 1,, e n R n. End R (R n ) f, a 1j a 2j f(e j ) =., (a 1j, a 2j,, a nj R, j = 1, 2,, n) a nj. M n (R) a f a f = (a ij ) 1 i,j n, f(u) = a f u (u R n ). f a f, End R (R n ) = M n (R)., End R (R n ) M n (R). 2.3 D. 1 M n (D). 5

6 e ij, (1 i, j n) M n (D). I M n (D) {0 Mn (D)}, 0 Mn (D) a I. a (k, l) a kl 0. (a 1 kl e ik) a e li = e ii i = 1, 2,, n. I 1 Mn (D) = e 11 + e e nn = n (a 1 kl e ik) a e li I i=1, I. I = M n (D). 2 D n M n (D). D n e 1, e 2,, e n, M n (D) {e ij } 1 i,j n. D n 0 u = e 1 a e n a n, (a 1,, a n D). j a j 0, i = 1, 2,, n, a 1 j e ij u a 1 j e ij u = e i., e 1, e 2,, e n u M n (D) M n (D)u. M n (D)u = D n u D, u 0. 3 M n (D) D n. M n (D) D n. M n (D) {e ij } 1 i,j n, 1 Mn (D) = e 11 + e e nn. M M n (D), 0 u M. u = 1 Mn (D)u = e 11 u + e 22 u + + e nn u, i e ii u 0. e ii u M n (D) M n (D)e ii u. M M = M n (D)e ii u. M n (D)e ii M n (D) D n. M n (D)e ii M n (D)e ii u = M : a au M n (D), M n (D)e ii = D n., M n (D). M M n (D). M n (D) {a1 Mn (D) a D} D, M D. M n (D) = 1 i,j n De ij 6

7 , M D. M D. M M n (D),., D, M, M M n (D). N. u M u N. N, M/N, M = N + M n (D)u. M n (D)u = M n (D)e 11 u + M n (D)e 22 u + + M n (D)e nn u, i, {0} M n (D)e ii u N., D n = M n (D)e ii u, M n (D)e ii u. M n (D)e ii u N M n (D)e ii u M n (D), M n (D)e ii u N M n (D)e ii u N = {0}. M/N M = N + M n (D)e ii u = N M n (D)e ii u = N D n. D. 7

8 3 R, I {0} R. A R, B R AB = { a i b i a i A, b i B}. 3.1 R J, {0} J I, I. I R. End R (I): R I 1 I. (1) x I, x 0, I = xr. (2) x I, xi {0}, I e, e 2 = e xe = x. (1) x 0, {0} xr I. I I = xr. (2) R φ : I I φ(a) = xa. φ End R (I), φ Aut R (I). x I, I e φ(e) = x. xe = x. x(e 2 e) = 0, e 2 = e., R 0 e e 2 = e. e, ere = {eae a R}, R., eae, ebe ere eae + ebe = e(a + b)e ere (eae)(ebe) = e(aeb)e ere. e(eae) = eae = (eae)e, e ere, 0 R ere. ere. 2 e R, e I = er. R I End R (I). f End R (I), ψ(f) = f(e)e, ψ(f) ere, ψ : End R (I) ere. I, ψ. f End R (I), f(e) I = er f(e)e Ie = ere. ψ. f, g End R (I). ψ(f + g) = (f + g)(e)e = (f(e) + g(e))e = f(e)e + g(e)e = ψ(f) + ψ(g) 8

9 . f(e) = ea, g(e) = eb, f R e 2 = e ψ(f g) = (f g)(e)e = f(g(e))e = f(eb)e = f(e)be = f(e 2 )be = f(e)ebe = eaebe = (eae)(ebe) = ψ(f)ψ(g). id I End R (I) I, ψ(id I ) = e 2 = e. ψ.. c R, f c End R (I) f c (x) = ecx (x I). ψ(f c ) = ece ψ. Kerψ End R (I). I End R (I), Kerψ = {0}. ψ. 3.2 Wedderburn I, R I S = End R (I), S I T = End S (I). I (S, R) (S, T ). R a, ι a T u ι a = ua (u I). a ι a ι : R T. 3 a I, t T ι a t = ι a t. ι I ι(i) T. I b, s b S s b u = bu (u I)., t T a I (ba) t = (s b a) t = s b (a t) = b(a t), b (ι a t) = (b ι a ) t = b ι a t b I. ι a t = ι a t. 4 R, ι : R T. R, ι. ι. ι ι(r). ι 1R = 1 T ι(r), ι(r)t ι(r). I R RI. I, RI. R R = RI., ι(r) = ι(r)ι(i), ι(i)t = ι(i) ι(r)t ι(r). 2 R. (1) (2). 9

10 (1) R. (2) R D n M n (D). (1). I. R I, End R (I). D = End R (I). I D. J = {a R Ia = {0}}, J R. J R, R J = {0}. II {0}, I e. I I = er, D Ie = ere. Ie D. S = {a R dim D Ia < }, S R, e S, {0} S. R S = R, 1 R S, dim D I = n <. R = End D (I) = M n (D). 5 D 1 D 2 M n1 (D 1 ) = M n2 (D 2 ), n 1 = n 2 D 1 = D2. R = M n1 (D 1 ) = M n2 (D 2 ), I. D = End R (I). e 11 M n1 (D 1 ), I = e 11 M n1 (D 1 ). e 11 I,, D = e 11 Re 11 = D1. D = D 2 D 1 = D2., dim D I = n 1 = n 2. 6 R Z R. R Z R a Z R, a 0, Ra = ar, Ra. Ra = R, a a 1 R. b R, a b = b a, b a 1 = a 1 b. a 1 Z R. 7 D. Z Mn (D) = Z D. 10

11 4 K. 4.1 K R,. (CS1) R K. (CS2) Z R = K.,. A(K) = K A s (K) = K A cs (K) = K D(K) = K D(K) A cs (K) A s (K) A(K).. 3 R A s (K), K D R = M n (D). R A cs (K) D D(K). 4 K. A s (K) = A cs (K), D(K) = {K}, A A s (K) M n (K). A A s (K), K Z A. Z A /K K = Z A, A A cs (K). A s (K) = A cs (K)., A = M n (D), D D(K). a D, K K(a) D, K(a)/K, K = K(a), a K. D = K, A = M n (K). 4.2 A, B A(K), A B K A B. A B A(K). 8 A, B A(K) = Z A B = Z A Z B. {b i } B K, A B x x = i x i b i, (x i A) 11

12 . x Z A B, a A (a 1)x = x(a 1), (ax i ) b i = (x i a) b i i i. i ax i = x i a, x i Z A. Z A B Z A B. {a i } Z A K, Z A B y y = a i y i, (y i B) i. y Z A B y i Z B, Z A B Z A Z B. Z A Z B Z A B. 5 A A cs (K), B A(K). 0 I A B, I 1 A b 0. A, 0 a A, x 1, x 1, x 2, x 2,, x m, x m A. x I m x i ax i = 1 A (2) i=1 p(x) = min{j x = a 1 b a j b j, a i A, b i B} p = min{p(x) 0 x I}. p = 1. p > 1. x I, p(x) = p, x = a 1 b a p b p. a p = 1 A., a p 1 a p. ( p(x) < p ). a p 1 / Z A, c A, ca p 1 a p 1 c 0 x = (c 1)x x(c 1) = (ca 1 a 1 c) b (ca p 1 a p 1 c) b p 1 0 p(x ) < p. p = 1. 0 a b I. (x i 1)(a b)(x i 1) = 1 A b I. i 12

13 1 A A cs (K), B A s (K) = A B A s (K). 0 I A B, 0 1 A b I. B, y 1, y 1,, y n, y n B n y i by i = 1 B. n (1 A y i )(1 A b)(1 A y i) = 1 A 1 B I. i=1 i=1 2 A, B A cs (K) = A B A cs (K) L/K. A A cs (K) A L A cs (L). 3 A A cs (K), dim K A. K/K K. A K = M n (K), dim K A = n 2. A A cs (K), D D(K), A = M m (D). deg A = dim K A A, inda = dim K D A. 4.4 A A(K). A : A A A a b = ba (a, b A). A K. A, A. A A cs (K) A A cs (K). 10 A A cs (K), deg A = n. A A = End K (A) = M n 2(K). 13

14 . a A b A, ψ a,b End K (A) ψ a,b (x) = axb (x A). ψ : A A End K (A) : m a i b i i=1 m i=1 ψ ai,b i, ψ. A A ψ. ψ. 4.5 Brauer A, B A cs (K), 1 p, q Z A M p (K) = B M q (K), A B Brauer, A B.. A [A].. Br(K) = A cs (K)/ = 11 D 1, D 2 D(K). M n (D 1 ) M m (D 2 ) D 1 D 2 D 1 = D A, B A cs (K) A = B A B deg A = deg B 13 D(K)/ = K.. D(K)/ = Br(K) : (D ) [D]. 14

15 14 Br(K) [A] [B] = [A B], (A, B A cs (K)). (1) well-definied. (2) [K]. (3) [A] [A] 1 = [A ]... Br(K). K Brauer. 15 L/K.. Br(K) Br(L) : [A] [A L] 15

16 5 5.1 A A(K), X A, {a A x X ax = xa} X A Z A (X). Z A (X) A. 6 A A cs (K), B A. Z A (B) dim A = dim B dim Z A (B). ψ : A A End K (A), A A A. B A A A, A B A. f End K (A), f End B A (A) f ψ b,a = ψ b,a f (b B, a A )., ψ. Z A A (B A ) = End B A (A) Z A A (B A ) = Z A (B) Z A = ZA (B). B A, I B A, D = End B A (I), I D s B A = End D (I) = M s (D)., B A A = I k. Z A (B) = End B A (I k ) = M k (D), Z A (B). dim Z A (B) = k 2 dim D, dim B A = s 2 dim D, dim A = k dim I = ks dim D. dim B dim C = s k k2 dim D = dim A. 4 A A cs (K), B A K, K A = B Z A (B). Z A (B) K. B Z A (B) K. B Z A (B) A : x y xy B Z A (B) = A. 16

17 5.2 A A cs (K). L/K L A A. A. 16 L A. (1) C = Z A (L), C Z C = L. C A cs (L). (2) deg A = [L : K] deg C. [L : K] deg A. (1) C. B = L, D C = Z A (L) = M k (D), L A = Ms (D). L A A cs (L) D D(L). C A cs (L). (2) dim L C = m 2, dim K A = dim K L dim K C = [L : K]m 2 [L : K] = [L : K] 2 m 2. deg A = [L : K]m. L A, [L : K] = deg(a).. M n (C). 17 D D(K). L D, L. D. C = Z D (L)., L = Z C C D. L C, c C, c L. C, L(c) L. L = C, deg D = [L : K] 5.3 A A cs (K), deg A = n. L/K, L A. A L = M n (L) 17

18 18 A A cs (K), D D(K), A D. L/K L A L D 19 D D(K). L/K D deg D [L : K]. deg D = d, D L = M d (L). V D L. V = L d, D M d (L) = V d. [L : K] = dim D D L = dim D M d (L) = dim D V d = d dim D V, d [L : K]. 20 D D(K), L D.. L L L D deg D = d. ( =) [L : K] d, d [L : K]. [L : K] = d. (= ) [L : K] = d dim L D = d. D (D, L). φ : D L = D L End L (D) = M d (L). D L.. 4 D D(K), D K D = K. d = deg D, L D. [L : K] = d. (ch(k) = 0 ) 1 < d L/K. d = 1, D = K. (ch(k) = p > 0 ) L/K d = [L : K] = p m. K K, D K = M d (K)., K D D K : a a 1 φ : D M d (K). a D, K(a)/K a pk = λ K. p k = [K(a) : K] d = p m., M d (K) (φ(a) λ 1/pk 1 d ) pk = 0 18

19 . φ(a) λ 1/pk 1 d, 1 < d, p d dλ 1/pk = 0, Tr(φ(a)) dλ 1/pk = Tr(φ(a) λ 1/pk 1 d ) = 0. Tr(φ(a)) = 0 (a D) M d (L) = D K = φ(d)k, x M d (K) Tr(x) = 0. d = 1, D = K. 7 D D(K), L D, L/K. L = K D. C = Z D (L), C D(L). L C, L L C, L /K, L. L = C, L = Z D (L), L. 5 A A cs (K), K. 19

20 6 Skolem-Noether 6.1 Skolem-Noether 5 R A s (K), M 1 M 2 R. dim K M 1 = dim K M 2 M 1 = M2., M 1, M 2 R N. M 1 = N k, M 2 = N l dim K M 1 = dim K M 2 k = l M 1 = M2. A A cs (K). a A, i a : A A : i a (x) = axa 1 (x A). i a K.. Inn(A) = {i a a A }.. a i a A /Z A = Inn(A). 8 (Skolem-Noether) A A cs (K), R A s (K), f, g : R A K. i a Inn(A), g = i a f. D D(K), A = M n (D) = End D (D n ). D D. V = D n ( D ) (A D ) V V : (a d, v) avd A D. f : R A V R D V f. V f = V, (R D ) V f V f : (α d, v) f(α)vd. g : R A V R D V g. R D A s (K). dim K V f = dim K V = dim K V g, R D V f = Vg. φ : V f V g, φ(f(α)vd) = g(α)φ(v)d (α R, d D, v V ). D φ Aut D (V ) = A. A a a f(α)v = φ(f(α)v) = g(α)φ(v) = g(α)a v v V. g(α) = af(α)a 1 = i a f(α) α R, g = i a f. 20

21 6 A A cs (K), R A. f : R A, i a Inn(A) f = i a R. 6.2 Brauer 9 (Frobenius) D D(R) = D = H. 1 < deg D. L D. 1 < d = [L : R] = 2. L = C. τ : C C : α + β 1 α β 1. Skolem-Noether, i a Inn(D), i a C = τ. a(α + β 1)a 1 = α β 1. i a 2 C, a 2 Z D (C) = C. τ(a 2 ) = aa 2 a 1 = a 2 a 2 R. a 2 > 0 a 2 = t 2 (t R), a = ±t, i a C τ. a 2 < 0, a 2 = t 2., i = 1, j = at 1, k = ij i 2 = j 2 = k 2 = ijk = 1, dim R D = 4 D = R1 + Ri + Rj + Rk, D = H. 7 Br(R) = {±1}. 6.3 Brauer 6 G, H G. ghg 1 G. g G [G : H] = n > 1, H = h G = nh. G/H g 1,, g n ghg 1 = g G n i=1 g i Hg 1 i ghg 1 g i Hgi 1 (n 1) = nh (n 1) < nh = G g G i=1 21

22 10 (Wedderburn). D. K = Z D. q = K, deg D = d > 1. D L 1, L 2, [L 1 : K] = [L 2 : K] = d L 1 = L 2 = q d. L 1 = L2. Skolem-Noether, D. L 1 L 2. D. D,, D = al 1 a 1 a D,,. 8 K Br(K) = 0. 22

23 7 7.1 K. L/K n, Γ L/K =< σ >. a K, x. x 1 = x 0, x, x 2,, x n 1 L n V = L1 + Lx + + Lx n 1. (αx i )(βx j ) = { (ασ i (β))x i+j (i + j < n) (aασ i (β))x i+j n (i + j n) (0 i, j n 1. α, β L).. (CY1) x n = a. (CY2) xα = σ(α)x (α L). V K., V = (L/K, σ, a). 21 (L/K, σ, a) A cs (K).. 22 L A A cs (K), u A. (CY0) 1, u,, u n 1 L A. (CY1) u n K. (CY2) uαu 1 = σ(α) (α L). A = (L/K, σ, u n ). Skolem-Noether, σ : L L i u Inn(A). uαu 1 = σ(α) u n αu n = σ n (α) = α u n Z A (L) = L, (α L). (α L). σ(u n ) = uu n u 1 = u n u n K. 1, u,, u n 1 L. dim L A = n, A. 23

24 9 A A cs (K), A, A K. 11 (1) n k (L/K, σ, a) = (L/K, σ k, a k ). (2) (L/K, σ, a) = (L/K, σ, b) ab 1 Nr L/K (L ). (3) (L/K, σ, a) = M n (K) a Nr L/K (L ). (1) (L/K, σ k, a k ) = L1 + Ly + + Ly n 1. φ : (L/K, σ k, a k ) (L/K, σ, a) : { φ(α) = α (α L) φ(y) = x k. (2) A = (L/K, σ, a), B = (L/K, σ, b). ( =) ab 1 = δ δ σ δ σn 1 B = L1 + Ly + + Ly n 1. φ : A B : { φ(α) = α (α L) φ(x) = δy. (= ) φ : A B. Inn(B), φ L = id. i x Inn(A), i y Inn(B), i φ(x) L = i y L = σ. φ(x)y 1 Z B (L) = L, δ L φ(x) = δy. a = φ(a) = φ(x n ) = (δy) n = Nr L/K (δ)y n = Nr L/K (δ)b. (3) (2) (L/K, σ, a) = (L/K, σ, 1) a Nr L/K (L ), (L/K, σ, 1) = M n (K). M n (K) = End K (L). φ : (L/K, σ, 1) End K (L) : { φ(α) = α (α L) φ(x) = σ. 23 a, b K. (L/K, σ, a) (L/K, σ, b) (L/K, σ, ab). 24

25 7.2 n ch(k), µ n K 1 n. µ n K. 1 n ζ K. L/K n, Kummer, a K, L = K( n a), Γ L/K µ n : σ σ( n a) n a. Γ L/K σ σ( n a) n a = ζ. b K, (L/K, σ, b) (L/K, σ, b) = L1 + Ly + + Ly n 1 { (CY1) y n = b (CY2) yα = σ(α)y (α L). x = n a L = K1 + Kx + + Kx n 1, x n = a., (CY2) α = x. yα = σ(α)y (α L) yx = σ(x)y = ζxy. (L/K, σ, b) (L/K, σ, b) = Kx i y j 05i,j5n 1 x n = a y n = b yx = ζxy. a K. µ n K, n ζ µ n. a, b K, x n = a Kx i y j y n = b yx = ζxy 05i,j5n 1 K (a, b) n, ( ) a, b. K, ζ 24 (a, b) n A cs (K). A = (a, b) n, K(x) = K1 + Kx + + Kx n 1. K(x) A,. A u = f 0 (x)1 + f 1 (x)y + + f n 1 (x)y n 1, f j (x) = c 0j 1 + c 1j x + + c n 1j x n 1 K(x) 25

26 . ( ) u Z A. yu = uy c ij = c ij ζ i (0 i, j n 1). i 0 c ij = 0. u = c c 01 y + + c 0n 1 y n 1. xu = ux c 01 = = c 0n 1 = 0, u = c 00 K. Z A = K. ( ) 0 I A. l = min{deg y u u I}, I l = {u I deg y u = l} I, k = min{deg f(x) f(x) M}, M = { u I l y l } K(x). u I l u = f(x)y l + v + f 0 (x)1, deg f(x) = k, 1 deg y v l 1, f 0 (x) K(x). f(x) = c 0 + c 1 x + + c k x k. 1 k. I l u yuy 1 = (f(x) f(ζx))y l +, f(x) f(ζx) M 0 I x 1 (u yuy 1 ) = x 1 (f(x) f(ζx))y l +, deg x 1 (f(x) f(ζx)) = k 1, k. k = 0, f(x) = c 0 K u = c 0 y l + v + f 0 (x) I l. 1 l. f 0 (x) = 0, uy 1 I, deg y uy 1 = l 1 l, f 0 (x) 0. I u ζ l xux 1 = v ζ l xvx 1 + (1 ζ l )f 0 (x) 0 deg y (u ζ l xux 1 ) l 1, l. l = 0, I u = c 0 K I = A. 7 A A cs (K), deg A = n, A = M n (K) A K n = K K. 26

27 (= ). ( =). K n e 1,, e n e e n = 1, e i e j = δ ij e j. A A A = e 1 A e n A. A = M m (D), D D(K), D m A, m n. m deg A = n, m = n D = K. 12 d n. a, b K ( a d ) ( ), b a, b = K, ζ K, ζ d M d (K). (a n, b) n = Mn (K). n = dm. ζ d 1 m. A = (a d, b) n = Kx i y j. K(x) = K1 + Kx + + Kx n 1, K. K(x) = K[X]/(X n a d ) d 1 = K[X]/(X m aζ im ) i=0 (ζ i X mod (X m aζ im )) i=0,1,,d 1 K(x) z. z m = a, y d zy d = ζz. z w = y d ( ) B = Kz i w j a, b = K, ζ d 05i,j5m 1. B A cs (K), A = B Z A (B). x m Z A (B), t = x m Z A (B) K(t) = K1 + Kt + + Kt d 1 = K[X]/(X d a d ) = K d. deg Z A (B) = d, Z A (B) = M d (K). 27

28 10 a, b K, L X n a = 0 m = [L : K]. σ Γ L/K σ( n a) = ζ n/m n a, (a, b) n = (L/K, σ, b) Mn/q (K). (a, b) n = Mn (K) b Nr L/K (L ). c = ( n a) m K, L = K( m c). a = c n/m ( ) ( ) (a, b) n c, b c, b = K, ζ n/m M n/m (K), = K, ζ n/m (L/K, σ, b).,. (a, b) n = Mn (K) (L/K, σ, b) = M m (K) b Nr L/K (L ) 13 a, b, c K. (1) (a, b) n = (b 1, a) n (2) (a, b) n (a, c) n (a, bc) n, (a, b) n (c, b) n (ac, b) n. (3) (a, 1 a) n = Mn (K). (1) (a, b) n = Kx i y j, (b 1, a) n ) = Kz i w j. φ : (a, b) n (b 1, a) n : φ(x) = w, φ(y) = w 1. (2) L X n a = 0, (a, b) n (a, c) n (L/K, σ, b) (L/K, σ, c) (L/K, σ, bc) (a, bc) n. (3) (a, 1 a) n = Kx i y j. z = x + y z n = (x + y) n = x n + y n = a + 1 a = 1. K(z) = K1 + Kz + + Kz n 1 = K n, (a, 1 a) n = Mn (K). 28

29 7.3 K. A A cs (K) deg A = 2,. A. 8 A A cs (K). (1) A, A = M 2 (K). (2) A A. (1) A, inda < deg A = 2 A = M 2 (K). (2) A, A K L. [L : K] = deg A = 2, L/K. A. A = (L/K, σ, b). ch(k) 2, L = K( a) A = (a, b) 2 = K1 + Kx + Ky + Kxy, x 2 = a, y 2 = b, yx = xy. ch(k) = 2, L X 2 + X + a = 0,. A = K1 + Kx + Ky + Kxy, x 2 + x = a, y 2 = b, yxy 1 = x

30 8 8.1 Noether A A cs (K),. (CP) A K L. L/K Γ = Γ L/K. σ Γ Skolem-Noether, u σ A σ = i uσ L. ξ σ,τ = u σ u τ u 1 στ Z A (L) = L (σ, τ Γ). u σ u τ = ξ σ,τ u στ (u σ u τ )u ρ = u σ (u τ u ρ ) ξ σ,τ ξ στ,ρ = ξ σ,τρ σ(ξ τ,ρ ) (σ, τ, ρ Γ). A B = σ Γ Lu σ, {u σ } σ Γ L, dim K B = [L : K] 2 = dim K A, A A = σ Γ Lu σ, { uσ αu 1 σ = σ(α) (α L) u σ u τ = ξ σ,τ u στ. L/K, Γ = Γ L/K. ξ : Γ Γ L : (σ, τ) ξ σ,τ, ξ Noether 2. (NF) ξ σ,τ ξ στ,ρ = ξ σ,τρ σ(ξ τ,ρ ) (σ, τ, ρ Γ). Noether Z 2 (Γ, L ). 9 ξ Z 2 (Γ, L ) ξ e,σ = ξ e,e, ξ σ,e = σ(ξ e,e ), (σ Γ). (NF), σ = τ = e, ξ e,e ξ e,ρ = ξe,ρ 2, ξ e,e = ξ e,ρ. τ = ρ = e, ξσ,e 2 = ξ σ,e σ(ξ e,e ), ξ σ,e = σ(ξ e,e ). 30

31 10 ξ, η Z 2 (Γ, L ), ξ η : Γ Γ L (ξ η) σ,τ = ξ σ,τ η σ,τ, (σ, τ Γ), ξ η Z 2 (Γ, L ), Z 2 (Γ, L ). 1 1 σ,τ = 1 (σ, τ Γ). t : Γ L, t : Γ Γ L t σ,τ = t σ σ(t τ )t 1 στ, (σ, τ Γ), t.. B 2 (Γ, L ). 11 B 2 (Γ, L ) Z 2 (Γ, L ). Z 2 (Γ, L )/B 2 (Γ, L ), Γ L, H 2 (Γ, L ). ξ Z 2 (Γ, L ) [ξ], [1]. 8.2 L/K, Γ = Γ L/K. V = σ Γ Lx σ {x σ } σ Γ L. ξ Z 2 (Γ, L ), V (CP1) x σ α = σ(α)x σ, (α L, σ Γ). (CP2) x σ x τ = ξ σ,τ x στ, (σ, τ Γ).. V ξ 1 e,e x e K. L Γ ξ, (L, Γ, ξ).. 25 (L, Γ, ξ) A cs (K), L (L, Γ, ξ). 31

32 A = (L, Γ, ξ). ( ) Z A z = σ Γ a σ x σ. α L, αz = zα αa σ = σ(α)a σ, (σ Γ). σ e σ(α) α α L, a σ = 0. z = a e x e, zx σ = x σ z σ(a e )ξ σ,e = a e ξ e,σ. σ(a e ξ e,e ) = a e ξ e,e, a e ξ e,e K. z = a e x e = a e ξ e,e 1 A K. ( ) I A. π : A A/I. π L : L A/I, L A/I. x σ = π(x σ ). x σ α = σ(α)x σ (α L, σ Γ). {x σ } σ Γ L., J Γ {x σ } σ J. J Γ, τ J x τ = σ J a σ x σ. α L τ(α)x τ = x τ α = σ J a σ x σ α = σ J a σ σ(α)x σ 0 = σ J a σ (τ(α) σ(α))x σ. {x σ } σ J σ J a σ 0, a σ (τ(α) σ(α)) = 0. τ(α) = σ(α), (α L), τ = σ J. J = Γ. dim L A/I = dim L A = [L : K], I = A A cs (K), A K. 32

33 14 ξ, η Z 2 (Γ, L ) (L, Γ, ξ) = (L, Γ, η) [ξ] = [η]. A = (L, Γ, ξ) = Lx σ, B = (L, Γ, η) = Ly σ. (= ) φ : A B. Skolem-Noether φ L = id. φ(x σ )α = σ(α)φ(x σ ), (α L). φ(x σ ) = c τ y τ τ Γ,. φ(x σ ) = c σ y σ,. φ(x σ x τ ) = c σ y σ c τ y τ = c σ σ(c τ )y σ y τ = c σ σ(c τ )η σ,τ y στ φ(x σ x τ ) = φ(ξ σ,τ x στ ) = ξ σ,τ c στ y στ ξ σ,τ = ĉ σ,τ η σ,τ, [ξ] = [η]. ( =) [ξ] = [η], c : Γ L ξ = ĉ η. φ : A B φ( a σ x σ ) = a σ c σ y σ,. 15 (L, Γ, 1) = M n (K). (L, Γ, 1) = Lx σ. φ : (L, Γ, 1) End K (L) : { φ(α) = α (α L) φ(x σ ) = σ (σ Γ). 12 (L, Γ, ξ) = M n (K) [ξ] = [1]. 16 ξ, η Z 2 (Γ, L ) (L, Γ, ξ) (L, Γ, η) (L, Γ, ξ η). 33

34 A = (L, Γ, ξ) = Lx σ, B = (L, Γ, η) = Ly σ, C = (L, Γ, ξ η) = Lz σ. A, B L. A L, L V = A L B. V (A, B), A B. C V. (a L b )(a b) = a a L b b = a a L b b, (a, a A, b, b B) C V V : αz σ (a L b) = a (αx σ ) L y σ b = αx σ a L y σ b αz σ βz τ (a L b) = αz σ (βx τ a L y τ b) = (αx σ βx τ a) L (y σ y τ b) = (ασ(β)ξ σ,τ x στ a) L (η σ,τ y στ b) = (ασ(β)ξ σ,τ η σ,τ z στ )(a L b) = (αz σ βz τ )(a L b), V C. V (C, A B), K (A B) End C (V ). (A B),. dim K V = [L : K] dim L A dim L B = n 3 = n dim K C, C V = C n. End C (V ) = End C (C n ) = M n (End C (C)) = M n (C ) = C M n (K). dim K End C (V ) = n 2 dim K C = n 4 = dim K (A B) (A B) = EndC (V ) = C M n (K). A B C. 34

35 8.3 L/K n, Γ = Γ L/K =< σ >. 26 a K, (L/K, σ, a). ξ σ i,σ j = { 1 (i + j < n) a (i + j n) ξ, (L/K, σ, a) = (L, Γ, ξ). (L/K, σ, a) L,. (L/K, σ, a) = Lxσ, Noether ξ. 27 ξ Z 2 (Γ, L ), (L, Γ, ξ).,,. 35

36 9 Brauer 9.1 L/K,, Br(K) Br(L) : [A] [A L]. Br(L/K). Br(L/K) = {[A] Br(A) [A L] = [1]}. 28 Br(K) = Br(L/K). L/K :, A A cs (K) K. 17 L/K,. H 2 (Γ, L ) Br(L/K) : [ξ] [(L, Γ, ξ)] Z 2 (Γ, L ) Br(K) : ξ [(L, Γ, ξ)], B 2 (Γ, L ). H 2 (Γ, L ) Br(K). Br(L/K). D D(K) [D] Br(L/K). deg D = d, [D ] = [D] 1 Br(L/K) L D = Md (L). V = L d L D. V (L, D). D V n. End D (V ) = M n (D) dim K V = d[l : K] = n dim K D = nd 2 dim K End D (V ) = dim K M n (D) = d 2 n 2 = (dn) 2 = [L : K] 2. L End D (V ) : α α, L End D (V ), L End D (V )., End D (V ) = (L, Γ, ξ) ξ Z 2 (Γ, L ).. [D] = [M n (D)] = [End D (V )] = [(L, Γ, ξ)] 36

37 9.2 A A cs (K), [A] Br(K) A, exp(a). 29 m = ind(a) [A] m = [1]. exp(a) ind(a). L/K [A] Br(L/K). [A] = [(L, Γ, ξ)]. [A] m = [(L, Γ, ξ m )], ξ m B 2 (Γ, L ). B = (L, Γ, ξ) = M r (D), D D(K), V = D r B. L B, V L. [L : K] = deg(l, Γ, ξ) = deg M r (D) = rm dim K V = r dim K D = rm 2 = dim L V [L : K] = dim L V rm dim L V = m. V L v 1,, v m, b B bv i = b i1 v b in v n m m (b ij ) M m (L)., x σ X σ. ξ m B 2 (Γ, L ). B = Lx σ ξ σ,τ X στ = σ(x τ )X σ ξ m σ,τ det(x στ ) = σ(det(x τ )) det(x σ ) 13 Br(K). 30 p ind(a), p exp(a).. [A] = [(L, Γ, ξ)], (L, Γ, ξ) = M r (D) d = ind(a) = deg(d) Γ = [L : K] = (deg M r (D)) = dr p Γ. Γ p-sylow Γ p. L p = L Γp [L : L p ] = Γ p. p [L p : K], L p D. A. exp(a L p ) 1. [A L p ] Br(L/L p ), ind(a L p ) [L : L p ] = Γ p. exp(a L p ) ind(a L p ) p exp(a L p ). exp(a L p ) exp(a) p exp(a). 37

38 D 1, D 2 D(K), deg(d 1 ) deg(d 2 ) D 1 D 2. d 1 = deg(d 1 ), d 2 = deg(d 2 ). D 1 D 2 = Mn (D), D D(K). n = 1. D 3 D(K) D 1 D = M r (D 3 ), M d 2 1 (D 2 ) = Md 2 1 (K) D 2 = D 1 D 1 D 2 = D 1 M n (D) = M nr (D 3 ), D 2 = D3 d 2 1 = nr. n d2 1. n d2 2 n = D D(K), d = deg(d) d = p e 1 1 pe 2 2 pe l l D 1,, D l D(K). D = D 1 D 2 D l, deg(d i ) = p e i i, i = 1, 2, l. D 1,, D l. d = d 1 d 2, d 1 d 2,, D 1, D 2 D(K) D = D 1 D 2, deg(d 1 ) = d 1, deg(d 2 ) = d 2,. q 1, q 2 Z d 1 q 1 + d 2 q 2 = 1. Br(K), [D] d 2q 2 = [D 1 ], [D] d 1q 1 = [D 2 ] D 1, D 2 D(K). D 1, D 2. [D 1 D 2 ] = [D] d 2q 2 +d 1 q 1 = [D]. deg(d) = ind(d) = d, [D 1 ] d 1 = [D] d 1d 2 q 2 = [D] dq 2 = [1]. exp(d 1 ) d 1. exp(d 2 ) d 2. d 1 d 2 exp(d 1 ) exp(d 2 ). ind(d 1 ) = deg(d 1 ) ind(d 2 ) = deg(d 2 ). D 1 D 2, [D] = [D 1 D 2 ] D = D 1 D , Brauer.. 1,. 38

39 19 (Wedderburn, Albert, Dickson) D D(K), deg D {2, 3, 4, 6, 12} D. 20 (Frobenius, Hasse, Tsen, Wedderburn, Witt) K. C, C(t), C((t)). R, R(t), R((t)).. ().. D D(K).,. 21 (Brussel) K Q Q(t) Q Q((t)). D(K),. 2 D D(K) deg D = p, D. p = 2, 3. p = 5. D D(K), D = D 1 D 2 K D 1, K D 2 D(K) decomposable, indecomposable., D indecomposable, deg(d)., p p e p d, exp(d) = p e, deg(d) = p d indecomposable. 22 (Albert, Jacob) (p e, p d ) (2, 4), K indecomposable D D(K) exp(d) = p e, deg(d) = p d.,, exp(d) = 2, deg(d) = 4,. 39

40 10 Merkurjev-Suslin 10.1 Milnor K K K K 1 (K). l : K K 1 (K) : a l(a). l(1) = 0, l(a n ) = nl(a), l(ab) = l(a) + l(b) (n Z, a, b K ). K 1 (K) Z, TK 1 (K). TK 1 (K) = T n K 1 (K), n=0 T 0 K 1 (K) = Z, T n K 1 (K) = K 1 (K) Z Z K 1 (K). TK 1 (K) J = {l(a) l(1 a) a K, a 1}, TK 1 (K)/J. π : TK 1 (K) TK 1 (K)/J, K n (K) = π(t n K 1 (K)), {a 1,, a n } = π(l(a 1 ) l(a n )).. TK 1 (K)/J = K n (K) n=0 K n (K) K Milnor K. K 2 (K), (M1) {ab, c} = {a, c} + {b, c}, {a, bc} = {a, b} + {a, c}. (M2) {a, 1 a} = 0, (a 1). (M3) {a, 1} = {1, a} = 0, {a, b 1 } = {a, b}.. 31 a, b K. (M4) {a, a} = 0. (M5) {a, b} = {b, a}. 40

41 10.2 Merkurjev-Suslin n. Br(K) n = {[A] Br(K) [A] n = [1]} 23 (Merkurjev-Suslin) (n, ch(k)) = 1, µ n K. 1 n ζ µ n. ( ) a, b K 2 (K) Br(K) : {a, b} (a, b) n = K, ζ well-definied, nk 2 (K), Br(K) n.. K 2 (K)/nK 2 (K) = Br(K) n 14 (n, ch(k)) = 1 µ n K. A A cs (K) exp(a) n, a 1, b 1,, a r, b r K. A (a 1, b 1 ) n (a r, b r ) n 15 ch(k) 2. A A cs (K), exp(a) = 2, a 1, b 1,, a r, b r K. A (a 1, b 1 ) 2 (a r, b r ) 2 41

42 11 Brauer 11.1 D. D. v : D R v. (VA1) v(x) 0, (x D), v(x) = 0 x = 0. (VA2) v(xy) = v(x)v(y), (x, y D). (VA3) v(x + y) max(v(x), v(y)), (x, y D)., v(x) = 1, (x D ). v D, O v = {x D v(x) 1}. v. O v., O v, O v = {x D v(x) = 1} P v = {x O v v(x) < 1}., O v /P v v. v, D δ v δ v (x, y) = v(x y), (x, y D). D. δ v D D v. D v, v D v. D = D v v. v w D. r > 0, w = v r, v w, v w.., v w O v O w P v P w (D, δ v ) (D, δ w ) v(d ) R v. ϖ O v, v(d ) = {v(ϖ) n n Z}. ϖ O v. v. D O v v, O v /P v 42

43 v D (D ). O D, P D, f D = O D /P D. K v, K D K. v D w, D. e(d/k) = [w(d ) : v(k )], f(d/k) = [f D : f K ]. [D : K] = e(d/k)f(d/k). K v, L/K. L. e(l/k) = 1, L/K. n, K n /K, [K n : K] = n. q = [O v : P v ], K n = K(µ q n 1). K n /K. Frobenius K, K n /K Γ Kn/K, f Kn /f K Γ fkn /f K. Γ Kn/K Γ fkn /f K : σ σ OKn. f K = Fq, f Kn = Fq n, Γ fk n /f K σ : f Kn f Kn : x x q. σ Γ Kn /K σ n, K n /K Frobenius Brauer 24 K, D D(K), deg D = d. D d K d /K e(d/k) = f(d/k) = [K d, K]. K d D, D. 25 K, ϖ K. θ K : Q/Z Br(K) : m n mod Z [(K n /K, σ n, ϖ m )]. 43

44 12 Brauer 12.1 K/Q. K V K,f. V K,f. V K,f v. v V K,f, K v K δ v. K v., θ v : Br(K v ) Q/Z. K C Hom(K, C). Hom(K, C) φ ψ φ(x) = ψ(x) (x K), V K,. V K, K. w V K,, w ψ w Hom(K, C). K w = ψ w (K) C. ψ w, K w = R K w = C. K w = R w, K w = C w. V K,1, V K,2. [K : Q] = (V K,1 ) + 2 (V K,2 ). w V K,1, θ w : Br(K w ) = Br(R) 1 2 Z/Z : θ w([a]) =. V K = V K, V K,f. V K K. { 1/2 + Z ([A] = [H]) 0 + Z ([A] = [1]) 12.2 K/Q. A A cs (K) v V K A v = A K K v. ([A v ]) v VK v V K Br(K v ) 44

45 ,. Br(K) v V K Br(K v ) : [A] ([A v ]) v VK 26 (Albert-Hasse-Brauer-Noether) A A cs (K), deg A = n. (1) v V K, A v = Mn (K v ). ([A v ]) v VK v V K Br(K v ). (2) v V K A v = Mn (K v ) A = M n (K). Br(K) v V K Br(K v ). (3). 1 Br(K) v V K Br(K v ) Br(K) = (α v) v V K Q/Z P θv Q/Z 1 α v = 0, α w 1 2 Z/Z (w V K,1), α w = 0 (w V K,2 ). v V K 27 A A cs (K) ind(a) = exp(a). 45

46 13 Hausdorff-Banach-Tarski Paradox Brauer-Severi 46

47 , Bourbaki ( ) van der Waerden ( III).., (associative ring), [1] Rowen, Ring Theory I & II, Academic Press, 1988 [2] Pierce, Associative Algebras, Springer Verlag, [1],. [2], [1]. Wedderburn,,, [1],, [2]. [3] Herstein, Noncommutative Rings, The Mathematical Association of America, 1968., Brauer, [4] Draxl, Skew Fields, Cambridge University Press, 1983 [5] Farb & Dennis, Noncommutative Algebra, Springer Verlag, 1993 [6] Jacobson, Finite-Dimensional Division Algebras over Fields, Springer Verlag, 1996 [7] Kersten, Brauergruppen von Körpern, Vieweg, Brauer, [1],[2],[4],[5], [8] Deuring, Algebren, Springer Verlag, 1968 [9] Albert, Structure of Algebras, American Mathematical Society, 1961 [10] Weil, Basic Number Theory, Springer Verlag, [11] Saltman, Lectures on Division Algebras, American Mathematical Society, [12] Platonov & Yanchevskii, Finite-Dimensional Division Algebras, in Algebra IX, Encyclopaedia of Mathematical Sciences, Springer Verlag, [12]., (order). [13] Reiner, Maximal Orders (2nd ed.), Clarendon Press, [14] Vignéra, Arithmétique des Algèbres de Quaternions, Springer Verlag, Eichler [15],,,, 21, [16] Kleinert, Units in Skew Fields, Birkhäuser, 2000., (involution), [17] Knus, Merkurjev, Rost & Tignol, The Book of Involutions, American Mathematical Society,