1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +

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1 ALGEBRA II Hiroshi SUZUKI Department of Mathematics International Christian University [email protected]

2 1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac + bc (distributive law)) R4 R 0 R 1 1x = x1 = x x R R5 (commutative ring) R5 ab = ba for all a, b R R1 R3 R4 (unital ring) R R2 R4 U(R) 0x = x0 = 0 ( 1) 0 U(R) R {0} R {0} R # (Pf.) 0 = 0x + ( 0x) = (0 + 0)x + ( 0x) = 0x + 0x + ( 0x) = 0x + 0 = 0x 1.2 U(R) = R {0} = R # (skew field) R5 (field) 1.3 R a b 0 ab = 0 [ba = 0] a (left zero divisor) [ (right zero divisor)] (zero divisor) 0 R6 ab = 0 a = 0 or b = 0 R1 R Z 1 1

3 2. Q R C 3. R R Mat n (R) 4. n Z n = Z/nZ = { 0, 1,..., n 1} n n Z R x f(x) = a 0 + a 1 x + + a n x n, a i R for i = 0, 1,..., n x R x R[x] x R f = f(x) = a 0 + a 1 x + + a n x n, g = g(x) = b 0 + b 1 x + + b m x m R[x] f + g = i fg = i (a i + b i )x i ( ) a j b i j x i j R[x] R f = f(x) = a 0 + a 1 x + + a n x n R[x], a n 0 n = deg f f f(x) = 0 deg f = deg 0 = 1.1 R f, g R[x] (1) deg(f + g) max(deg f, deg g) (2) deg(fg) = deg f + deg g R (1) (2) fg = 0 (2) = deg fg = deg f + deg g. deg f = deg g = f = 0 g = R f, g R[x] g R q, r R[x] deg r < deg g f = gq + r R q, r R[x] 1 2

4 f = a n x n + + a 1 x + a 0 a n 0 g = b m x m + + b 1 x + b 0 n < m q = 0 r = f n m n = deg f b m h = f (a n b 1 m )x n m g f deg h < n R[x] q 1, r deg r < deg g h = gq 1 +r f = g(q 1 + (a n b 1 m )x n m ) + r q = q 1 + (a n b 1 m )x n m R f = gq + r = gq + r, deg r, deg r < deg g g(q q ) = r r deg g + deg(q q ) = deg(g(q q )) = deg(r r) max(deg r, deg r) < deg g. g 0 q q = 0 r r = 0 q = q r = r n R[x 1,..., x n ] = (R[x 1,..., x n 1 ])[x n ] a i1,...,i n x i 1 1 x in n, a i1,...,i n R. i 1,...,i n R[x, y] = (R[x])[y] = (R[y])[x] 1 3

5 2 R I R/I R/I xy (x + I)(y + I) (x + I)(y + I) xy + I x = 0 y = 0 xi I Iy I 2.1 R I a, b I a + b I a I, r R ra I, [ar I]. I R [ ] A B R A + B = {a + b a A, b B}. AB = { i a i b i a i A, b B} R I, J I J I + J 2. I, J R IJ IJ I J I R I R R/I (a + I) + (b + I) = (a + b) + I, (a + I) (b + I) = (ab) + I R/I (quotient ring) a R Ra [ar] [ ] (principal) [ ] R Ra = ar (a) 2 1

6 0 = {0} R R R I R [ ] I = R U(R) I. (Pf.) I = R 1 I U(R) U(R) I u U(R) I r R R I I = R r = r(u 1 u) = (ru 1 )u RI I. 2.1 R R R [ ] 0 R ( ) I 0 R a I {0} a U(R) I = R ( ) a 0 a Ra Ra 0 1 R = Ra R b ba = 1 b 0 R = Rb R c cb = 1 c = c1 = c(ba) = (cb)a = 1a = a ab = ba = 1 R 0 R X (well-ordered set) (PID : principal ideal domain) 2. R (well-ordered set) X ρ : R X R (Euclidean domain) (a) 0 a R ρ(0) < ρ(a). (b) a, b R (a 0) b = aq + r, ρ(r) < ρ(a) q, r R 2.2 R I R I = 0 I 0 {ρ(x) 0 x I} X ρ(a) a I b I b = aq + r ρ(r) < ρ(a) q, r R r = b aq I a r = 0 b Ra b I = Ra 2 2

7 ρ : Z {0} N ρ(a) = a Z 2.2 Z Z Z 2. K ρ : K[x] {, 0} N ρ(f) = deg f 1.2 K[x] 2.2 K[x] 2 3

8 3 3.1 R R f : R R f(a + b) = f(a) + f(b), f(ab) = f(a)f(b), f(1 R ) = f R f R R ((ring) homomorophism) f R R 3.1 I R (R I) 3.2 R S f : R R/I, (a a + I) a, b S a b S, ab S, 1 R S S R (subring) R S (extension ring) f : R R (1) Kerf = {a R f(a) = 0} (2) Imf = {f(a) a R} R R R f : R R R/Kerf Imf. 3.1 R/Kerf Imf f : R/Kerf Imf, (a + Kerf f(a)) well-defined f((a + Kerf)(b + Kerf)) = f(ab + Kerf) = f(ab) = f(a)f(b) = f(a + Kerf) f(b + Kerf) f(1 R/Kerf ) = f(1 + Kerf) = f(1 R ) = 1 R. f f(a) = f(b) f(a b) = 0 a b Kerf a + Kerf = b + Kerf f well-defined 3 1

9 3.2 K L α L φ : K[x] L (f(x) f(α)) Imφ K[α] K[α] = {f(α) f(x) K[x]} K[x]/Kerφ K[α] Kerφ K[x] p(x) K[x] Kerφ = K[x]p(x) = (p(x)) p(x) = 0 Kerφ = 0 α K p(x) 0 p(x) 1 p(x) Kerφ α K p(x) α K K = Q L = C e π Q Q (transcendental number) α = 1 Kerφ = (x 2 + 1) Q[ 1] = {a + b 1 a, b Q} α = 3 2 Kerφ = (x 3 2) Q[ 3 2] = {a + b c( 3 2) 2 a, b Q} 3.3 A Mat n (C) ψ : C[x] Mat n (C) (f(x) f(a)) Hamilton-Cayley Kerψ (det(xi A)) 0 monic Kerψ = p A (x) p A (x) p A det(xi A) A = ( ), B = ( ), C = ( p A (x) = (x 1)(x + 2), p B (x) = (x 2) 2, p C (x) = x 2 ) 3 2

10 4 R I R/I I 4.1 (1) R I( R) ab I a I b I I (prime ideal) (2) R I( R) I J : R I = J J = R I (maximal ideal) 4.1 R I (1) R/I I (2) R/I I (3) (1) R/I ā, b R/I, ā b = 0 ā = 0 b = 0 a, b R, (a + I)(b + I) = ab + I = I a + I = I b + I = I a, b R, ab I a I b I (x + I = y + I x y I ) (2) 2.1 R/I R/I 0 R/I R J I R I R (3) I R/I R/I I R (0) R (0) R 4.2 R (PID) I R I (0) I I 4 1

11 4.1 I = (a) (0) I J I R R J = (b) a (a) = I J = (b) a = bc c R I = (a) b I c I b I J = (b) (a) = I J I c I = (a) c = ad d R a = bc = bad = abd a(bd 1) = 0 a 0 bd = 1 R = (1) (b) = J J = R I 4.3 n Z (n) (n) n (n) (m) m n (m) = (n) m = ±n (m) (m) (n) (n) = (m) (n) = (1) n ±n ±1 (1) = Z (n) n Z 4.2 Z n = Z/(n) n 4.2 R 1 f(x) R[x] f(x) = g(x)h(x) deg g > 0 deg h > 0 R R 4.4 K f(x) K[x] (f(x)) (f(x)) f(x) f(x) deg f(x) 1 K[x] (f(x)) f(x) 4 2

12 f(x) f(x) = g(x)h(x) deg g(x) > 1 deg h(x) > 1 (f(x)) (g(x)) K[x] U(K[x]) = U(K) = K (f 1 (x)) = (f 2 (x)) f 1 (x) = cf 2 (x) (c K ). (f(x)) (g(x)) (f(x)) K[x] f(x) = g(x)h(x) f(x) 4.1 x 2 +1 x 2 2 Q (x 2 +1) (x 2 2) Q[x] Q[ 1] Q[x]/(x 2 +1) Q[ 2] Q[x]/(x 2 2) 4.5 [Eisenstein ] p f(x) = a n x n + + a 1 x + a 0 Z[x] a n 0 (mod p), a n 1 a 1 a 0 0 (mod p), a 0 0 (mod p 2 ) f(x) Z f(x) = g(x)h(x), r = deg g > 0, s = deg h > 0, g(x) = b r x r + + b 0, h(x) = c s x s + + c 0 a 0 = b 0 c 0 p p 2 p b 0 c 0 a n = b r c s p c s p c 0 p i c i p c 0 c 1 c i 1 0 c i (mod p). a i = b 0 c i + b 1 c i b i c 0 b 0 c i 0 (mod p). n = i < s = deg h f(x) Z 7.7 Q x n 2 x 3 3x 2 9x 6 Z Q 4 3

13 5 5.1 R 1, R 2,..., R n R = R 1 R 2 R n = {(a 1, a 2,..., a n ) a i R i, i = 1,..., n} (a 1,..., a n ) + (b 1,..., b n ) = (a 1 + b 1,..., a n + b n ) (a 1,..., a n ) (b 1,..., b n ) = (a 1 b 1,..., a n b n ) R R R 1,..., R n R = R 1 R 2 R n. 1 R = (1 R1, 1 R2,..., 1 Rn ) 0 R = (0 R1, 0 R2,..., 0 Rn ) Ri = {(0,..., 0, a, 0,..., 0) a R i } i 0 Ri R R I, J I + J = R I J I + J = R x + y = 1 x I, y J Z (m) (n) (m, n) = 1 m n 1 x + y = 1 x (m), y (n) am + bn = 1 a, b Z (m, n) = (Chinise Remainer s Theorem) 5.1 [ ] R I 1, I 2,..., I n i j I i + I j = R a 1, a 2,..., a n R x a i (mod I i ) i = 1, 2,..., n x R n = 2 1 = c 1 + c 2 c 1 I 1, c 2 I 2 x = a 1 c 2 + a 2 c 1 (mod I 1 ) x a 1 c 2 + a 2 c 1 a 1 c 2 a 1 (1 c 1 ) a 1 a 1 c 1 a 1 5 1

14 x a 2 (mod I 2 ) n > 2 i x i R x i 1 (mod I i ), j i x i 0 (mod I j ). i = 1 j 2 I 1 + I j = R c (j) 1 + c j = 1 c (j) 1 I 1 c j I j n 1 = (c (j) 1 + c j ) c 2 c n (mod I 1 ) j=2 1 c 2 c n = c 1 c 1 I 1 R = I 1 + I 2 I n I 1 I 2 I n n = 2 x 1 R x 1 1 (mod I 1 ), x 1 0 (mod I 2 I n ) j 2 I 2 I n I j x 1 0 (mod I j ) i x i x = a 1 x a n x n x a 1 x a n x n (mod I i ) a i x i (mod I i ) a i (mod I i ) 5 2

15 6 6.1 R S (i) a, b S ab S (ii) 1 S, 0 S S R (multiplicative subset) R 2. P R R P R S R S (a, s) (a, s ) (as a s)t = 0 t S (a, s) a/s S 1 R S 1 R (a 1 /s 1 ) + (a 2 /s 2 ) = (a 1 s 2 + a 2 s 1 )/s 1 s 2 (a 1 /s 1 )(a 2 /s 2 ) = (a 1 a 2 /s 1 s 2 ) S 1 R R S (quotient ring) 1. 0 S 1 R = 0/1 1 S 1 R = 1/1 (a/s) = ( a)/s s S s/1 U(S 1 R) 2. S a/s = a /s as a s = 0 3. φ S : R S 1 R (a a/1) φ S S S R S 1 R R (ring of total quotients) 2. R R (quotient field) Q(R) (a) Q(Z) = Q 6 1

16 (b) Q(K[x 1,..., x n ]) = K(x 1,..., x n ) = {f/g f, g K[x 1,... x n ], g 0} (c) P R (R P ) 1 R R P R P (localization) 6.2 R M R (local ring) (0) R M I R Zorn I R I M M R 6.1 R [(1) (2) ] (1) R (2) R U(R) R (1) (2) M R M R M U(R) = M R U(R) a R U(R) Ra R a Ra M R U(R) M M = R U(R) (2) (1) J R J R I = R U(R) J U(R) = J I I R 6.2 P R R P P = {a/s a P, s P } P R P (Pf.) (a/s) + (b/t) = (at + bs)/st a, b P, s, t P at + bs P st P (at + bs)/st P r R (r/t)(a/s) = ar/ts P P R P a/s P a P. (Pf.) ( ) a P a/s P ( ) a/s P a/s = a /s a P s P (as a s)t = 0 t P as t = a st P a P 6 2

17 R P P = U(R P ) (Pf.) ( ) a/s P a P s/a R P a/s U(R P ) ( ) 1 P 1/1 P a/s U(R P ) P a P (a/s)(b/t) = 1/1 b R t P t P abt = stt P P U(R P ) P = R P P = U(R P ) 6 3

18 7 7.1 R a, b R (a) (b) a = bc c R b a (a) = (b) a = bu u U(R) a b R p 0 p = uv u U(R) v U(R) p 7.1 R R (UFD = Unique Factorization Domain) (i) a R a = p 1 p 2 p r (p i ) (ii) a = p 1 p 2 p r = q 1 q 2 q s (p i, q j r = s p i q i 7.1 R 0 p R (1) (p) p (2) R UFD (p) p (1) p = ab a (p) b (p) a (p) (a) (p) = (ab) (a) a p p = au u U(R) a(b u) = p p = 0 R b = u U(R) (2) p ab (p) a b U(R) ab = pc a = p 1 p r b = q 1 q s c = v 1 v t p 1 p r q 1 q s = pv 1 v t p p i p q j p p i a = p 1 p r (p i ) = (p) p q j b = q 1 q s (q j ) = (p) R 7.2 R p 0 (1) p 7 1

19 (2) (p) (3) (p) 4.2 (2) (3) 7.1 (2) (1) (1) (3) (p) I = (q) R p = qa q a (q) = R (p) = (q) (p) 7.3 R 0 a R U(R) (a) R (a) (p 1 ) 7.2 p 1 (a) (p 1 ) a = p 1 a 1 p 1 U(R) (a 1 ) (a) a 1 U(R) p 2 a 1 = p 2 a 2 (a = p 1 p 2 a 2 ) a i (a) (a 1 ) (a 2 ) (a i ) i=1 (a i ) R i=1 (a i ) = (d) i d (a i ) (a i ) = (a i+1 ) r a r p r a r a = p 1 p 2 (p r a r ) a = p 1 p 2 p r = q 1 q 2 q s r s r q 1 q 2 q s = a (p 1 ) (p 1 ) q i (p 1 ) i (q i ) (p 1 ) q i p 1 q 1 = p 1 u u U(R) p 1 p 2 p r = p 1 uq 2 q s p 2 p r = uq 2 q s r = s p i q i Z[x] Q[x 1,, x n ] 7.1 Z[ 5] = {a + b 5 a, b Z} 2 (2) α = a + b 5 N(α) = αᾱ = a 2 + 5b 2 α U(Z[ 5]) N(α) = 1 α = ±1. (Pf.) ±1 U(Z[ 5]) αβ = 1 1 = N(αβ) = N(α)N(β) a 2 + 5b 2 = N(α) = 1 a, b Z b = 0 a = ±1 7 2

20 2 (Pf.) 2 = αβ N(α) 1 N(β) 1 α = a + b 5 4 = N(2) = N(αβ) = N(α)N(β) a 2 + 5b 2 = N(α) = 2 N(α) = 1 N(β) = 1 α, β (2) (Pf.) (1 + 5)(1 5) = 6 (2) 1 ± 5 (2) 1 ± 5 = 2γ 6 = N(1 ± 5) = 4N(γ) 1 ± 5 (2) (2) 7 3

21 7.2 R K = Q(R) d a 1,..., a n R 1. d a i, i = 1, 2,..., n 2. c a i, i = 1, 2,..., n c d l a 1,..., a n R 1. a i l, i = 1, 2,..., n 2. a i m, i = 1, 2,..., n l m a 1, a 2,..., a n 1 a 1, a 2,..., a n (coprime) a 0, a 1,..., a n f(x) = a 0 + a 1 x + + a n x n R[x] (primitive polynomial) R R R = Z 4 6 ±2 7.4 R R[x 1, x 2,..., x n ] 7.5 f(x) K[x] c K f 0 (x) R[x] f(x) = cf 0 (x) c R I(f) f(x) = (b 0 /a 0 ) + (b 1 /a 1 )x + + (b n /a n )x n 0 a i, b j R m a 0, a 1,..., a n m = a i c i d b 0 c 0, b 1 c 1,..., b n c n de i = b i c i e 0, e 1,..., e n f(x) = (b 0 /a 0 ) + (b 1 /a 1 )x + + (b n /a n )x n = 1 m (b 0c 0 + b 1 c 1 x + + b n c n x n ) = d m (e 0 + e 1 x + + e n x n ) c = d/m f 0 (x) = e 0 + e 1 x + + e n x n f(x) = cf 0 (x) = c f 0(x) f 0 (x) f 0(x) R c = b/a c = b /a a b a b R a bf 0 (x) = ab f 0(x) 7 4

22 f 0 (x) f 0(x) a b = ab u u U(R) c = b/a = (b /a )u = c u K c, c c = cu u U(R) c c f(x) K[x] f(x) R[x] I(f) R f(x) I(f) (1) (2) f(x), g(x) K[x] I(fg) I(f)I(g) (1) f(x) = a 0 + a 1 x + + a l x l g(x) = b 0 + b 1 x + + b m x m h(x) = f(x)g(x) = c 0 + c 1 x + + c n x n, p c i, ; i = 0, 1,..., n p a i p i i 0 b j p j j 0 c i0 +j 0 = a 0 b i0 +j a i0 1b j a i0 b j0 + a i0 +1b j a i0 +j 0 b 0 a i0 b j0 (mod (p)) 0 (mod (p)) (2) f(x) = I(f)f 0 (x) g(x) = I(g)g 0 (x) f 0 (x) g 0 (x) f(x)g(x) = I(f)I(g)f 0 (x)g 0 (x) f 0 (x)g 0 (x) (1) I(f)I(g) I(fg) 7.7 f(x) R[x] f(x) R[x] K[x] f(x) K[x] R[x] K[x] f(x) = g(x)h(x) g(x), h(x) K[x] g(x) = I(g)g 0 (x) h(x) = I(h)h 0 (x) f 0 (x), g 0 (x) f(x) = I(g)I(h)g 0 (x)h 0 (x) f(x) R[x] I(g)I(h) I(gh) R deg g 0 = deg g = 0 deg h 0 = deg h = 0 K[x] 7.8 f(x) R[x] (i) deg f = 0 f R (ii) deg f > 0 f 7 5

23 U(R[x]) = U(R) (i), (ii) f(x) f = gh g, h U(R[x]) = U(R) f R deg f = 0 f R deg f > 0 f f = I(f)f 0 I(f) U(R) f (ii) 7.4 R[x 1,..., x n 1, x n ] = (R[x 1,..., x n 1 ])[x n ] n = 1 R R[x] 0 f(x) R[x] deg f deg f = 0 R 7.8 (i) R[x] deg f > 0 f = gh deg g > 0, deg h > 0 deg g < deg f deg h < deg f g h f f = I(f)f 0 f 0 f (ii) I(f) R R[x] f = p 1 p k f 1 f l = q 1 q m g 1 g n f p 1,..., p k, q 1,..., q m R f 1,..., f l, g 1,..., g n 1 f 1 f l g 1 g n 7.6 I(f) p 1 p k q 1 q m u U(R) up 1 p k = q 1 q m R K[x] uf 1 f l = g 1 g n c i f i = g i c i K I(g i ) = 1 c i R g i c i U(R) 7 6

24 8 8.1 R M R M M, (r, m) rm M R- R- r(x + y) = rx + ry, (r + s)x = rx + ry, (rs)x = r(sx), 1x = x (x, y M, r, s R) R- R R- N M R- N rx N r R x N RN N N R f : M M R- f(a + b) = f(a) + f(b), f(ra) = rf(a), (r R, a, b M) f(ra) = rf(a) f R Z- 2. R R- I R- R I R 3. K K- K M R- S M { } < U >= r i u i r i R, u i U i U R- 2. U < U M =< U > M R- u 1, u 2,..., u n M = Ru 1 + Ru Ru n 3. r 1 u 1 + r 2 u r n u n = 0 (r i R) r 1 = r 2 = = r n = 0 u 1, u 2,..., u n R- M U R- U R- M U R- 8 1

25 V K K- V K- 8.3 R R- A A R R- a, b A, r R (ra)b = a(rb) = r(ab) R R 2. G = {1 = u 1, u 2,..., u n } G R- R[G] = Ru 1 Ru n ( n ) n n α i u i β j u j = α i β j u i u j. i=1 j=1 i,j=1 G A = C[G] V A- g G φ(g) : V V, (v gv) φ(g) GL(V ) φ : G GL(V ), (g φ(g)) φ : G GL(V ) V A M R- M 0 M M 8.1 (Schur s Lemma) M N R- (1) f : M N R- 0 f (2) End R (M) M M End R (M) f R- Kerf Imf R- (1) f 0 Kerf M Imf 0 Kerf = 0 Imf = N f (2) (1) 8 2

26 R-( ) M R- [ ] M [ ] 2. R R-( ) [ ] R ( )- [ ] 3. M R- M 1 M 2 M i (M 1 M 2 M i ) n M n = M n+1 = M [ ] 9.1 R- M [ ] M [ ] R- M M 1 M 2 {M i i N} M n M n = M n+1 = S M 1 M 2 M i M i S M i M i+1, M i M i R- M (i) M (ii) M R- R- (i) (ii) N M R- S N R- R- S N 0 N N 0 x N N 0 Rx + N 0 N 0 N 0 N = N 0 N (ii) (i) M 1 M 2 M N = i M i R- N =< u 1, u 2,..., u n > N M m u 1, u 2,..., u n N M m M m+1 N. M 9.1 M

27 1 9.4 R R[x 1, x 2,..., x n ] n = 1 I R[x] I i = {r R f(x) = a i x i + + a 1 x + a 0 I a i = r } R f(x) = a i x i + + a 1 x + a 0 I xf(x) = a i x i a 1 x 2 + a 0 x I I 0 I 1 I 2 R 9.1 I r = I r+1 = r 9.2 I 0, I 1,..., I r a i1,..., a isi I i (i = 0, 1,..., r) R f ij a ij I i I r s i I = R[x]f ij (x). i=0 j=1 f = a m x m + + a 1 x + a 0 I m = deg f m = 0 f = a 0 I 0 = s 0 j=1 Ra 0j = s 0 j=1 Rf 0j m > 0 r < m e = m r r m e = 0 a m I m = I m e = a m = s m e j=1 c j a (m e)j s m e deg(f(x) x e j=1 s m e j=1 Ra (m e)j c j f (m e)j (x)) < deg f(x) f r i=0 si j=1 R[x]f ij (x) R[x] 9.2 R[x] S R s 1,..., s n S R {s 1,..., s n } S S {s 1,..., s n } R- S Z[x] x Z- Z[x] Z- Z + Zx R R- R R 9 2

2014 (2014/04/01)

2014 (2014/04/01) 1 5 1.1...................................... 5 1.2...................................... 7 1.3...................................... 8 1.4............................... 10 1.5 Zorn...........................