2, Steven Roman GTM [8]., [3].,.

Transcription

1 ( ) :

2 2, Steven Roman GTM [8]., [3].,.

3 (UFD) (PID) Möbius UFD , (Algebraic Extnsions) distinguished (Galois Correspondence) ? Lifting

4 distinguished class

5 1 1.1 N = {1, 2,... }, Z = { 2, 1, 0, 1, 2,... }, Q, R, C (P, ) (partially ordered set, poset), x, y P x y (i) x x ( reflexivity) (ii) x y y x x = y ( anti-symmetry) (iii) x y y z x z ( transtivity) S P a S (upper bound) resp. (lower bound) : x S : x a (resp. x a) a S (maximum element) resp. (minimum element) : a S a S (resp. ) a S (maximal element) resp. (minimul element) : a S x a (resp. x a) x P S = P (resp. ), 1 P (resp. 0 P ),, 1 (resp. 0) N, Z, Q, R S N S ( ) P P, P (Zorn ) P, {x λ } λ Λ P, {x λ } λ Λ (resp. ) x 1 (resp. x 0 ), x 1 {x λ } λ Λ (join)) (resp. x 0 {x λ } λ Λ (meet)) ), x 1 = x λ (resp. x 0 = x λ )., x 0, x 1 λ Λ λ Λ (1) x λ x 1 ( λ Λ) (resp. x 0 x λ ( λ Λ)) (2) x λ y ( λ Λ) = x 1 y (resp. x 0 x λ ( λ Λ) = y x 0 ) ( ) L 2 x, y {x, y}, L (lattice)., {x, y}, {x, y} x y, x y (, join), (, meet). (i) (x y) z = x (y z), (x y) z = x (y z) (associative raws) (ii) x y = y x, x y = y x (commutative laws) (iii) x x = x x = x (idempotent) (iv) x (x t) = x = x (x y) (absorption laws) 1, (iv) x y x y. 5

6 6, L, (i) (iv), s t = s s t = t s t.,,,., 2,,, L (complete lattice) ( (monoid)) (monoid). S (x, y) xy, (i) ( ) (xy)z = x(yz) (ii) ( 1 ) 1 S s.t. x : x1 = 1x = x 1. (iv) ( ) xy = yx, n x 1, x 2,..., x n C n. C 1 = 1, C 2 = 1, C 3 = 2, C 4 = 5,.... C n C n =. C n+1 = 1 n+1( 2n n ). n C i C n i i= ( ) G (iii) ( ) x G, y G s.t. xy = yx = 1, G (group). y x, x 1., xy x + y, x x ( ) R, 2 (, ),, (unitary commutative ring). (i) (ii) (iii). x(y + z) = xy + xz, (x + y)z = xz + yz x R x 1, x (unit). R R R, x 0 R, y 0 R xy = 0, x. R, R (integral domain).

7 *** ( ) *** R = Z[ 1] = Z + Z K, K = K \ {0}, K, (field) ( ) R, S R, (1) x, y S x y S (2) x, y S xy S (3) 1 R S, S R (subring) (R ). R (4) x 0 S x 1 S, S R ( ) R, a R, (1) a R (2) x R, y a xy a a R (ideal) = {0} R R ( ) R, a R R/a = {x + a x R} (x + a) + (y + a) = x + y + a, (x + a)(y + a) = xy + a, well-defined, R/a. R a. x + R = {x + y y R} R = Z, m (m) = mz = {mx x Z}. Z/(m) = Z/mZ, Z/(m) = m Z Z/12Z Z/mZ Z/mZ m R, R, f : R R, (1) f(x + y) = f(x) + f(y) (2) f(xy) = f(x)f(y) (3) f(1 R ) = 1 R ( ) 2, (homomorphism). f (resp. ), (resp. ),,, R R., R = R, (automorphism) R, R, f : R R, (1) f S R f(s) R. 2 (3),,.

8 8 (2) f a R f(a) R. (3) S R f 1 (S) R. (4) a R a = f 1 (a ) R., n = f 1 (0) R K R f : K R, f(x) = 0 ( x R) ( ) f R R, n = f 1 (0) f, f : R/n R, a + n f(a), R/n R a ι (ι I) (family), ι I a ι ( ) S R (S) = a a S S., S = {a 1,..., a s }, ({a 1,..., a s }) (a 1,..., a s ),., s = 1, (a) (pricipal ideal). { k } (S) = r i a i r i R, a i S, k N. i= ( ) a i (i = 1,..., s) ( s ) a i = {a a s a i a i } a a s (a 1,..., a s )., ( s ) s a i = s a i ( a 1 a s ). i= Z + 8Z = 2Z (a b)(a, b) ab a b. i=1 i=1 j i=1 a (i) j a(i) j a i R 1, p R,. (1) R/p (2) a, b p, ab p a p b p (3) R a, b, ab p a p b p (4) R a, b, a p b p ab p

9 *** ( ) *** 9 p (prime ideal). 3. (1) (2) ab p, R/p (a + p)(b + p) = p, R/p a + p = p b + p = p. a p b p. (2) (4) R a, b, a p b p a a \ p, b b \ p. (2) ab p ab p. (4) (3) (3) (1) R/p R 1, m R,. (1) a, m a R a = m a = R. ( m ) (2) R/m m (maximal ideal). (1) (2) a + m m a m m (a) + m, m (a) + m = R. r R, m m ra + m = 1. r + m a + m. (2) (1) m a R x a \ m. R/m x + m m y R (x + m)(y + m) = 1 + m. 1 a a = R R 1, S (0 S) R (i.e. a, b S ab S), a R s.t. a S =., R I = {b b a b S = }. p 1 p, p.. I. b ι (ι I) I, c = ι I b ι, c, c I. 4,., Zorn, p I. p. b p, c p, bc p p (b) + p, p (c) + p ((b) + p) S, ((c) + p) S, s 1 = r 1 b + p 1, s 2 = r 2 c + p 2 s 1, s 2 S, r 1, r 2 R, p 1, p 2 p., S s 1 s 2 = r 1 r 2 bc + r 1 bp 2 + r 2 cp 1 + p 1 p 2 p p S = R 1, a R, a R m.. S = {1} R 1 R m.. a = R 0,. 3 R. 4 a c. c S =.

10 10 (1) R (2) R 0, R R.. a 0 R. 0 a a 1. f a : R a, x xa R a,. a = R. R 0 R ( ) R, F f : R F. (1) f (2) F x = f(a)f(b) 1., (F, f), (F, f ) (1) (2), φ : F F f = φ f.. F = {(a, b) a, b R, b 0}, F (a, b) = (c, d) ad = bc. F = F /, F (a, b) + (c, d) = (ad + bc, bd), (a, b)(c, d) = (ac, bd), well-defined, 0 F = (0 R, 1 R ), 1 F = (1 R, 1 R ). f : R F f(a) = (a, 1 R ) R, F R (field of quotients)., R ( ) R, a, b R b = ac ( c R), a b, b a, a b. a R x R, x = ϵa ϵ, a x, x a. 5 a 1, a 2,..., a n R (n 2) d a i ( i) x a i ( i) x d d R a 1, a 2,..., a n.,., d, d a 1, a 2,..., a n d d. a 1, a 2,..., a n a 1, a 2,..., a n. a 1, a 2,..., a n R a i m ( i) a i x ( i) m x m a 1, a 2,..., a n.,. 5 x a a x x a.

11 *** ( ) *** a, x R, (i) x a x = ϵ x = ϵa ( ϵ ) (ii) x a x, x a (iii) (a) (x) R (x) = (a) (x) = R R, a R, x R, x a x = ϵ x = ϵa ( ϵ ), a (irreducible element). p R p ab p a p b, p R R,.. p R, p = ab ab (p), a (p) b (p). a (p), p a, p = ab a p a p, p. b (p). 1.5 (UFD) R, R a (a 0) a = p 1 p 2 p r, (Unique Factorization Domain) ,., a R a = p 1 p r = q 1 q s 2, r = s, p 1 q 1,..., p r q r.. r. r = 1, p 1 = q 1 q s, q 1 q s (p 1 ) q 1 (p 1 )., p 1 q 1, q 2 q s s = 1. r > 1, p 1 p r = q 1 q s q 1 q s (p 1 ),, p 1 q 1. q 1 = ϵ 1 p 1 (ϵ 1 ) p 1 p r = ϵ 1 p 1 q s p 2 p r = ϵ 1 q 2 q s, r 1 = s 1, p 2 q 2,..., p r q r ,. (, ) R = Z[ 5] = {x + y 5 x, y Z}. w = x + y 5 R, w = x y 5, N(w) = ww = x y 2 Z N(w 1 w 2 ) = N(w 1 )N(w 2 ). (i) 2, 3, 1 ± 5 R. (ii) (2) R. (iii) (2, 1 + 5). (iv) R UFD. (6 = 2 3 = (1 + 5)(1 5) ). 1.6 (PID) R,, (Principal Integral Domain) ,. (, ) (p)

12 12., p (p) (p) p,, R, p R,. (i) (p) (ii) (p) R.. ( ) a 0 R,., a a = a 1 a 1 (a 1, a 1 ), a 1, a 1, 1., a 1, a 1 = a 2 a 2 (a 2, a 2 ), a 2, a 2, 1, a 2., (a) (a 1 ) (a 2 ) (a 3 ) R. i (a i ). 7 R i (a i ) = (b) b R., i 0 b (a i0 ), (a i0 ) = (a i0 +1) = (a i0 +2) =, R a, b. (i) a b. (ii) x, y R s.t. ax + by = 1.. (i) (ii) (a, b) = (c) c R c a c b c. (ii) (i) c a, b, c a, c b c 1 = ax + by c X, Y, R = Z[X, Y ] UFD. ( ) I = (X, Y ) R., R PID R 0 a v(a) 0,, Euclid (Euclidian Domain). (1) a, b R s.t. a 0, q, r such that b = aq + r, r = 0 v(r) < v(a) (2) a 0, b 0 v(a) v(ab) Euclid R.. a 0 R. S = {v(x) x a x 0} N v(a). a = (a)., x a x = aq + r r = 0 r 0 v(r) < v(a), r = x qa a, v(a).

13 *** ( ) *** : φ(n) n φ(n) Möbius n N, 1, 2,..., n n φ(n), Euler n N. φ(d) = n d n n = 12 φ(1) + φ(2) + φ(3) + φ(4) + φ(6) + φ(12) = = n = r i=1 pe i i n., φ(18) = φ(2 3 2 ) = 6. φ(n) = r i=1 p e i 1 i (p i 1) (Möbius ) n N µ(n). µ(1) = 1, n > 1, n = r i=1 pei i ( 1) r e 1 = = e r = 1, µ(n) = 0 otherwise. µ(n) n Möbius. n 1.8.2: µ(n) n µ(n) n Z, n > 1. µ(d) = 0 d n. n n = p e1 1 per r (r 1) µ (d) = d n µ (p x1 1 pxr 0 x 1 e 1,...,0 x r e r, x x r = x. r ) = 0 x 1 1,...,0 x r 1 ( 1) x1+ +xr = r ( ) r ( 1) r = (1 1) r = 0 x x=0 7.

14 n = 12 µ(1) + µ(2) + µ(3) + µ(4) + µ(6) + µ(12) = = ( ) f N, n N f(d) = g(n) d n.. d n ( n ) µ g(d) = f(n) d ( n ) µ g(d) = ( n ) µ f(d ) = d d d n d n d d d d n ( n ) µ f(d ) = f(d ) µ(t) d d n t n d,, n N. 1 a = 1, µ(t) = 0 a > 1. t a d n ( n ) µ d = f(n) d n = 12. µ(12) 1 + µ(6) 2 + µ(4) 3 + µ(3) 4 + µ(2) 6 + µ(1) 12 = = 4

15 R, R n X 1,..., X n f(x) = f(x 1,..., X n ) = a i1,...,i n X i1 X in i 1,...,i n 0 R[X 1,..., X n ], R n. ax i1 X in, i i n,. f, f(x), deg f. deg 0 = f(x, Y ) = 2 X 3 + X 2 Y + X Y Z[X, Y ] 3, g(x) = X X Q[X] 2., R. deg f 1 f R[X 1,..., X n ], f = gh, 1 deg g, deg h < f g, h R[X 1,..., X n ], f (degreewise reducible). f R[X 1,..., X n ] deg f 1,, (degreewise irreducible) h(x) = 2 X X X + 2 Z[X], g(x) = X X Q[X] f R[X 1,..., X n ] f R. f R[X 1,..., X n ] deg f = Z[X 1,..., X n ] ± F, f F [X] f F [X],, R, f R[X] R[X]. f(x) = 6(X 2 + X + 1) Z[X],, f(x) Z[X] 2, 3, X 2 + X + 1 Z[X]. Q[X] R[X] R 1. f R[X] f = a 0 + a 1 X + + a m X m = m a i X i (a i R, a m 0), m (degree), a m (leading coefficient). a m = 1 (monic) m., f, ia i X i 1 f, f (derivative). i= h(x) = 2 X X X + 2 Z[X] 3, 2 monic, h (X) = 6 X X + 2 Z[X] R R[X], f, g R[X], deg(fg) = deg f + deg g. f = m i=0 a ix i, g = n j=0 b jx j fg = m i=0 n j=0 a ib j X i+j a m b n i=0

16 R R[X 1,..., x n ], f, g R[X], deg(fg) = deg f + deg g. R[X 1,..., x n ] = R[X 1 ][X 2 ] [X n ] R, R[X 1,..., x n ] { } f(x1,..., X n ) R(X 1,..., X n ) = g(x 1,..., X n ) f(x 1,..., X n ), g(x 1,..., X n ) R[X 1,..., X n ], g(x 1,..., X n ) R, f, g R[X] g R f = gq + r, r = 0 deg f < deg g q, r R[X]..,. f = m i=0 a ix i, g = n j=0 b jx j, deg f < deg g. deg f = deg g,, q, r.. f = gq 1 + r 1 = gq 2 + r 2, deg r 1, deg r 2 < deg g g(q 1 q 2 ) = r 2 r 1. q 1 q 2 0 deg g(q 1 q 2 ) deg g > deg r 2 r 1. q 1 = q 2, r 1 = r R, f R[X], a R, q(x) R[X]., f(x) = (X a)q(x) + f(a), X a f(x) f(a) = 0. g(x) = X a f(x) = (X a)q(x) + r. X = a r = f(a) R m, f(x) R[X] m.. m. (i) m = 1,. (ii) m > 2, m 1. m, f(x) R[X] a f(x) = (X a)f 1 (X)., f 1 (X) m 1, m 1., f(x) m R, f(x) R[X] R a, f(a) = 0 f(x) = f R[X], a R, (X a) k f(x) (X a) k+1 f(x), a f(x) k, k f(x) a. 2,, R, a R f(x) R[X] k (k 2), a f (X) k 1. a R f(x) f(a) = f (a) = 0.., f(x) = (X a) k g(x), g(a) 0., f (X) = k(x a) k 1 g(x) + (x a)kg (X) = (X a) k 1 {kg(x) + (X a)g (X)} (X a) k 1 f(x).

17 *** ( ) *** F 1 F [X] Euclid F 1 F [X] PID F 1 f(x), g(x) F [X],. (i) f(x), g(x). (ii) f(x)u(x) + g(x)v(x) = 1 u(x), v(x) F [X] F 1 p(x) F [X],. (i) p(x) F [X] (ii) p(x) F [X] (iii) (p(x)) F [X] (iv) (p(x)) F [X] 2.3 UFD 1, R UFD., F R f(x) R[X], f(x) (primitive polynomial) h(x) = 2 X 3 +2 X 2 +2 X+2 Z[X] 2 primitive, k(x) = 6 X 2 +3 X+4 Z[X] primitive p R R p R[X], i.e., f(x), g(x) R[X] p f(x) p g(x) p f(x)g(x).. f(x) = a 0 + a 1 X + + a m X m, g(x) = b 0 + b 1 X + + b n X n (a m, b n 0). f(x), g(x) p, a j, b k, f(x)g(x) X j+k c j+k = a j+k b a j+1 b k 1 + a j b k + a j 1 b k a 0 b j+k. p a k b k, p, p c j+k p f(x)g(x) (Gauss s Lemma) f(x), g(x) R[X], f(x)g(x).., f(x)g(x), p f(x)g(x) p R p f(x) p g(x), f(x), g(x) f(x) = 2 X X + 3, g(x) = 2 X X + 2 Z[X], f(x)g(x) = 4 x x x x + 6 Z[X] a, b F, b = aϵ R ϵ R, a b 1, a b. 1 approx

18 R = Z, F = Q, Z = {±1}, 2 3 ± 2 3., R = Z[i] (i = 1), F = Q[i], Z = {±1, ±i}, 2 3 ± 2 3, ± 2 3 i f(x) F [X] f(x) = cg(x) c F g(x) R[X]. c, c = c(f) f(x) (content).. f(x) = a 0 + a 1 X + + a m X m, b 0, b 1,..., b m B b 0 b 1 b m f(x) = 1 ( a 0 B + a 1 B X + + a m B ) X m B b 0 b 1 b m, a i B b i R R A R, f(x) = A B (c 0 + c 1 X + + c m X m )., c = A B, f 0(X) = c 0 + c 1 X + + c m X m, f 0 (X) R[X]., f(x) = cf 0 (X) = c f 0(X) (f 0 (X), f 0(X) R[X] ), c = a b, c = a b ab f 0 (X) = a bf 0(X) R[X]., f 0 (X), f 0(X) ab a b. ab = a bϵ ϵ R., c = cϵ f 0 (X) = ϵf 0(X) R = Z, F = Q, g(x) = X X Q[X]. c(g) = 1 6, primitive k(x) = 6 X X + 4 Z[X], g(x) = c(g) k(x) f F [X], f R[X] c(f) R f R[X], f c(f) 1 f, g F [X], c(fg) = c(f)c(g) f(x), g(x) R[X], g(x), f(x) = g(x)h(x) h(x) F [X] h(x) R[X].. f(x) = g(x)h(x) R c(f) = c(g)c(h) = c(h), h(x) R[X] f(x) R[X], f(x) = g(x)h(x) g(x), h(x) F [X] g 1 (X), h 1 (X) R[X]. f(x) = g 1 (X)h 1 (X), deg g = deg g 1, deg h = deg h 1. g(x) = c(g)g 0 (X), h(x) = c(h)h 0 (X) (g 0, h 0 R[X] ) f(x) = c(g)c(h)g 0 (X)h 0 (X), 2.3.4, g 0 (X)h 0 (X), c(g)c(h) R.,, g 1 (X) = g 0 (X) R[X], h 1 (X) = c(g)c(h)h 0 (X) R[X] f(x) R[X], R[X] F [X] R R[X] f(x), (1) f(x) R[X] (2) f R (deg f = 0),, f (deg f 1).. (1) (2) f(x) R[X], f(x) = c(f)f 0 (X) (f 0 (X) R[X] ), f(x) c(f)f 0 (X) f(x) c(f) f(x) f 0 (X). f(x) c(f) R, f(x) f 0 (A) c(f) 1. (2) (1)

19 *** ( ) *** R R[X].. f(x) R[X], f(x) = c(f)f 0 (X) (c(f) R, f 0 (X) R[X] ). c(f) R c(f) = p 1 p r., R F F [X], F [X] f 0 (X) = g 1 (X) g s (X) (g i (X) F [X]) , h 1,..., h s R[X], f 0 (X) = h 1 (X) h s (X), deg g i = deg g i., f 0 1 c(h 1 ) c(h s ), h i F [X] R[X]., , f(x) = p 1 p r h 1 (X) h s (X) f R R[X 1,..., X n ].. R[X 1,..., X n ] = R[X 1,..., X n 1 ][X n ] (Eisenstein) R, p R, f(x) = a 0 + a 1 X + + a m X m p a m, p a i (i = 0,..., m 1), p 2 a 0 f(x) R[X].., f(x) = g(x)h(x), g(x) = b 0 + b 1 X + + b r X r, h(x) = c 0 + c 1 X + + c s X s, a 0 = b 0 c 0 p a 0, p 2 a 0, b 0, c 0 p. p b 0, p c 0 p a 1 = b 1 c 0 + b 0 c 1, p b 0, p c 0 p b 1. p a 2 = b 2 c 0 + b 1 c 1 + b 0 c 2, p b 0, p b 1, p c 0 p b 2. p b i (i = 1,..., m 1)., r < m p a m = b r c s p a m, r = m, s = f R[X], c, f(x) f(x + c) X 2 6 Q. ( 6.). p = 2 Eisenstein s Irreducibility Criterion p, f(x) = X p 1 + X p X + 1 Q.. f(x) = Xp 1 X 1 f(x + 1) = (X + 1)p 1 Eisenstein s Irreducibility Criterion F (X) = X Q. X = p i=1 ( ) p X i 1 i. f(x + 1) = X X X X + 2 p = 2 Eisenstein s Irreducibility Criterion.

20 f(x) = X 6 + X Q.. f(x + 1) = x x x x x x + 3 p = 3 Eisenstein s Irreducibility Criterion X 3 X 1 Q.. Z[X] f(x) = (X + b 1 X + b 0 )(X + c 0 ) 2 1 b 0 c 0 = 1 c 0 = ±1. f (±1) 0. m R, S, σ : R F. a R σ(a) a σ. f(x) = a i X i R[X] m a σ i X i S[X] f σ (X)., F R. i= R, F, σ : R F., f(x) R[X] 2, R. (1) deg f σ = deg f. (2) deg f σ F.. f(x) i=1 f(x) = g(x)h(x), 0 < deg g, deg h < deg f g, h R[X]., f σ (X) = g σ (X)h σ (X), deg g σ, deg h σ < deg f = deg f σ. f σ (X) R (PID), f(x) = a 0 + a 1 X + + a m X m R[X], p R p a m π p : R R/(p). f π p (X) R/(p), f(x) R f(x) = X X X + 1 Z[X] Z, p = 3 g(x) = f π 3 (X) = X X + 1 F 3 [X]. g(x) 3, 1, X = 0, 1, 2 0.

21 3 3.1, E, F E (1) F E. (2) x F x 1 F F E (subfield), E F (extension field)., E/F., F M E, F M, M E, M E/F (intermediate field), E/M/F., E i+1 /E i (tower). E 1 E 2 E 3 E n E M F 3.1.1: E/M/F F L, S L F (S) = {M L/M/F s.t. M S} S F, F S,, F S.,S S = {a 1,..., a n }, F (S) F (a 1,..., a n ), F (S) F., 1, F (a) F ( (simple extension)) F = Q, E = C, Q ( 2 ) F = Q. ( ) L/F. E λ (λ Λ) L, E λ F E λ,, E λ (λ Λ) (composite)., E, F L, E F EF. L/F, E λ E λ,. λ Λ λ Λ λ Λ λ Λ λ Λ E λ F = Q, L = C, E 1 = Q ( 2 ), E 2 = Q ( 3 ) L/F, E 1 E 2 = Q ( 2, 3 ), E 1 E 2 = E 1 E 2 = Q. E 1 E 2 E 1 E 2. 6 E 1 E E/F. (1) S 1, S 2 E F (S 1 S 2 ) = F (S 1 )(S 2 ) 21

22 22 Q ( 2, 3 ) Q ( 2 ) Q ( 3 ) Q ( 6 ) Q 3.1.2: E/M/F (2) {S λ } λ Λ S F (S) = λ Λ F (S λ ).. (1) F (S 1 S 2 ) F (S 1 ) F (S 1 S 2 ) S 2 F (S 1 S 2 ) F (S 1 )(S 2 )., F F (S 1 )(S 2 ) S 1 S 2 F (S 1 )(S 2 ) F (S 1 S 2 ) F (S 1 )(S 2 ). (2) S λ S F (S λ ) F (S)., F (S λ )., α, β F (S λ ) λ Λ s.t. λ Λ λ Λ λ Λ α, β F (S λ ), α ± β, αβ, α 1 F (S λ ). F (S λ ) F (S). λ Λ Q ( 2, 3 ) = Q ( 2 ) ( 3 ) E/F, S E { } f(a1,..., a n ) F (S) = g(a 1,..., a n ) n N, f, g F [X 1,..., X n ], a 1,..., a n S, g(a 1,..., a n ) 0., F S = { f(a1,..., a n ) g(a 1,..., a n ) } n N, f, g F [X 1,..., X n ], a 1,..., a n S, g(a 1,..., a n ) 0., F S E, F S S, F (S), F S F (S). F (S) = F S F S E Q ( 2 ),, x, y, z, w Z x + y 2 z + w 2 = (x + y 2)(z w 2) z 2 2w 2 = xz 2yw xw + yz z 2 + 2w2 z 2 2w 2 w 2 ( ) { Q 2 = a + b } 2 a, b Q (Lang) Cl 3 distinguished class. (1) (Tower Property) F K E K/F Cl E/K Cl E/F Cl (2) (Lifting Property) F E, F K K/F Cl EK/K Cl

23 *** ( ) *** 23 (1) E (2) EK (3) EK K E K E K F F F 3.1.3: Tower Property, Lifting Property, Closure under finite composites (3) (Closure under finite composites) F E, F K K/F Cl K/F Cl EK/F Cl F K E, E/F K/F... (.) distinguished class.. (1) (Tower Property) K = F (α 1,..., α m ), E = K(β 1,..., β n ), K = F (α 1,..., α m )(β 1,..., β n ) = F (α 1,..., α m, β 1,..., β n )., E/F, E = F (α 1,..., α m ), E = K(α 1,..., α m ), E/K. K/F, (2) (Lifting Property) K = F (α 1,..., α m ) S = {α 1,..., α m } EK = K(E) = F (S)(E) = F (E)(S) = E(S) = E(α 1,..., α m ). (3) (Closure under finite composites) K = F (α 1,..., α m ), E = F (β 1,..., β n ), EK = F (α 1,..., α m, β 1,..., β n ) F E, E F,,, E F,, [E; F ], E/F , 2 Q ( 2 ) Q, [Q ( 2 ) ; Q] = F K E {ω α } α A K/F, {η β } β B E/K, {ω α η β } α A, β B E/F. [E; F ] = [E; K] [K; F ].. {ω α η β } α A, β B F E.,, ξ E, K- ξ = µ β η β β B. µ β K µ β = λ α.β ω α (λ α.β F ), α A λ α.β ω α η β ξ = α A β B Q Q ( 2 ) Q ( 2, 3 ), 1, 2 Q ( 2 ) /Q, 1, 3 Q ( 2, 3 ) /Q ( 2 ) 1, 2, 3, 6 Q ( 2, 3 ) /Q. [Q ( 2, 3 ) ; Q] = 4.

24 F, n N (n 2) n 1 = n {}}{ = 0, n p, F (characteristic), ch(f ) = p. n, F 0, ch(f ) = F ch(f ) = p. (1) p > 0, p. (2) p > 0 F F p := Z/(p). (3) p = 0 F Q., F p Q (prime field).,.. (1) p > 0, p = ab a, b > 1, (a 1)(b 1) = a(b 1) = ab 1 = n 1 = 0, a 1 b 1 = 0, p p. (2) p > 0, φ : Z F n n 1, Ker φ = (p), φ(z) Z/(p) = F p. (3) p = 0, φ : Z F n n 1, φ : Q F F 2 = {0, 1} 2, : F F 3 = {0, 1, 2} 3, : F ch(f ) = p > 0, x, y F, n Z (n 0) (1) (x + y) pn = x pn + y pn, (xy) pn = x pn y pn. (2) φ : F F, x x pn.. (1) 1 p m, 0 < r < m, p ( ) m r. (2) (1) φ, φ(x) = x pn = 0 x = , p = 2, n = 2, p n = 4 ( ) ( ) 4 4 = 1, = 4, 0 1 ( ) 4 = 6, 2 ( ) 4 = 4, 3 ( ) 4 = 1 4

25 *** ( ) *** 25, F 2. (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 = x 4 + y F E, α E, f(α) = 0 0 f(x) F [X], F (algebraic). α, (transcendental) Q., Q. 2, 3 2, e, π. e π, π e. e + π, eπ E/F, α E (i) α F, F (α) F (X), F (α) F [X] F (X). (ii) α F, f 0 (α) = 0 0 monic F f 0 (X) F [X]. F (α) F [X]/(f(X))., F (α) = {f(α) f F [X]}, deg f 0 = n, F (α) F n.. (1) α F, ϕ : F (X) F (α) X α, F (X) F (α). (2) α F, φ : F [X] F (α) X α., m = Ker φ = φ 1 (0) 1 F [X]., f, g F [X] fg m f(α)g(α) = 0 f(α) = 0 g(α) = 0, f m g m. F [X], m. m = (f 0 (X)) monic f 0 (X) F [X] F [X]/(f 0 (X)) F (α). deg f 0 = n, F [X]/(f 0 (X)) a 0 + a 1 X + + a n 1 X n 1 + m, 1+m, X+m,..., X n 1 +m 1+m, X+m,..., X n 1 + m F., 1, α,..., α n 1 F (α) F, F (α); F ] = n, F (α) = {f(α) f F [X]} E/F, α E, (1) α F (2) F (α)/f. (1) (2) (ii) (2) (1), α F, (i) [F (α); F ] = E/F, α E α monic f 0 (X) α F (minimal polynomial) E/F, α E, f(x) F [X]

26 26 (1) f(x) F [X] α F (2) f(x) F [X] f(α) = 0 monic,. g(x) F [X], g(α) = 0 f(x) g(x) (3) f(x) F [X], g(α) = 0 monic g(x) F [X].. (1) (2) (ii) g(x) F [X], g(α) = 0 g(x) (f(x)) f(x) g(x) (2) (1) f 0 (X) F [X] (ii) f 0 (X) f(x), f(x) f 0 (X) f(x) f 0 (X). monic f(x) = f 0 (X). (2) (3) F = Q, E = C. α = 2 E F = Q f(x) = X 2 2 Q[X]. ( ) Q 2 Q[X]/(X 2 2) 2. Eisenstein, X 2 2 Q[X], m = (X 2 2), Q[X], f(x) = 0. ( 2 ) 2 = 2 Q ±X + m ( ) { Q 2 = a + b } 2 a, b Q, 1, 2 Q, Q ( 2 ) f(x) = : Q ( 2 ) X 2 2 = ( X 2 ) ( X + 2 ) ( ) { 3 Q 2 = a + b c 3 } 4 a, b, c Q Q[X]/(X 3 2) Q 3. 1, 3 2, 3 4 Q, Q ( 3 2 ) f(x) = X 3 2 = ( X 3 2 ) ( X X ) : n > 1, p, f(x) = x n p Z[X] , [Q ( n p ) ; Q] = n. Q ( n p) Q[X]/(X n p)

27 *** ( ) *** F = Q, E = C., f(x) = X 4 X 1 Q[X], Q 1, C 1 α K = Q(α) F 4. Q (α) = { a + bα + cα 2 + dα 3 a, b, c, d Q } Q[X]/(X 4 X 1) 1, α, α 2, α 4 Q. α 4 = α + 1, Q (α) f(x) f(x) = X 4 X 1 = (X α) ( X 3 + αx 2 + α 2 X + α 3 1 ). 1 α α 2 α α α 2 α 3 α α α 2 α 3 α + 1 α 2 α 2 α 3 α + 1 α 2 + α α 3 α 3 α + 1 α 2 + α α 3 + α : Q (α) (α X 4 X 1 ) F = F 2., f(x) = X 2 + X + 1 F 2 [X], F 2, 1 β K = F 2 (β) F 2. F 2 (β) = {a + bβ a, b F 2 } F 2 [X]/(X 2 + X + 1) 1, β F 2. a, b = 0, 1, K 2 2 = 4. β 2 = 1 + β, K = F 2 (β) f(x) β 1 + β β 1 + β β β β β 1 + β β 1 + β β β 1 + β β 1 + β β 0 β 1 + β β β 1 β 3.3.4: F 2 (β) (β X 2 + X + 1 ) f(x) = (X β)(x β 1) = (X β)(x β 2 ) F = F 2. f(x) = X 3 + X F 2 [X] 1 γ K = F 2 (γ) F 3. F 2 (γ) = { a + bγ + cγ 2 } a, b, c F 2 F2 [X]/(X 3 X 2 1) 1, γ, γ 2 F 2., a, b, c = 0, 1, K 2 3 = 8. γ 3 = 1 + γ 2, K = F 2 (γ) f(x) f(x) = (X γ)(x γ 2 )(X γ 2 γ 1) = (X γ)(x γ 2 )(X γ 4 ). 1 σ : Z Z/(2) f σ (X) = X 4 + X + 1 Z/(2)[2] F 2 [X] f(x) Z[X].

28 γ γ γ 1 + γ 2 γ + γ γ + γ γ γ γ 1 + γ 2 γ + γ γ + γ 2 γ 0 γ γ γ 2 γ + γ γ + γ γ γ 2 0 γ γ γ + γ γ γ γ + γ γ γ γ + γ γ 2 γ 1 + γ + γ 2 γ γ γ γ + γ 2 1 γ γ 2 γ γ γ + γ 2 0 γ + γ 2 1 γ 1 + γ + γ 2 γ γ 1 + γ γ + γ γ + γ γ γ + γ 2 γ γ 2 γ 3.3.5: F 2 (γ) (γ X 3 + X ) α = E = Q( 2, 3) Q(α) E. (1, 2, 3, 6) E Q, F α : E E, x αx (1, 2, 3, 6) A = A γ A (x) = x 4 10x 2 + 1, α 4 10α = 0. f(x) = X 4 10 X 2 +1 Z[X] Z, Q α Q. [Q(α); Q] = 4 Q( 2, 3) = Q(α) , t F, (i) F (t) t { } f(t) F (t) = f(x), g(x) F [X], g 0 g(t). F F (t)., s F (t) \ F t s = f(t) g(t) F (t)., f(t) g(t). p(x) = g(x)s f(x) F (s)[x], t p(x), t F (s),., 3.3.4, F (t) F (s),, F F (s)., F (s)/f., F (t)/f t F., F F (s). p(x) F (S). s F, (i) F (s) F (Y ). Y. F (s)[x] F (Y )[Y ] h(y, X) = g(x)y F (X) F (Y )[X]

29 *** ( ) *** 29 F (Y )., p(y, X) = g(x)y f(x) F (Y )., P (Y, X) = a(x){b(x)y + c(x)}, f(x) g(x). a(x) ) t F, F F (t). s = f(t) g(t) F (t), F (t) \ F., f(t) g(t)., F F (s) F (t),,,,. [F (t) : F (s)] = max(deg f, deg g) 2) t F, F (t) F F (t) F.. 1). (2) F K F (t) K F s ink \ F., F F (S) K F (t), 1) F (t)/f (S) 3.3.4, F (t)/k,. 3.4 (Algebraic Extnsions) F E, E F, E F ( ) (algebraic extnsion). E F, (transcedental extension) E/F [E : F ] = n α E 1, α, α 2,..., α n F a 0 + a 1 α + + a n α n = 0 0 a 0, a 1,..., a n F E/F,. (1) E/F (2) E/F (3) E/F,, E F S = {α 1,..., α n } E = {f(α 1,..., α n ) f(x 1,..., X n ) F [X 1,..., X n ] }.. (1) (2) 3.4.2,. (2) (3) (3) (1) E = F (α 1,..., α n ) = F (α 1 )... (α n ), F (α 1 )/F, α 2 F F (α 1 ), F (α 1, α 2 )/F (α 1 )., E/F.

30 30, (ii) n F = Q, E = Q ( 2, 3 ), 2, 3 Q E/F. α = 2, β = 3, α 2 = 2 Q, β 2 = 3 Q ( ) E = Q 2, 3 = { f(α, β) f(x, Y ) Q[X, Y ] } = { a+bα+cβ+dαβ a, b, c, d Q } = { a+b 2+c 3+d 6 a, b, c, d Q }., γ = 2 + 3, γ 4 10 γ = 0 E = Q(γ) E = Q (γ) = { f(γ) f(x) Q[X] } = { a + bγ + cγ 2 + dγ 3 a, b, c, d Q } L/F, S L ( ), S F., K = F (S) F, K = F (S) S F -. K = { f(α 1,..., α r ) α 1,..., α r S, f(x 1,..., X r ) F [X 1,..., X r ], r 1 }. α F (S) (2), T = {α 1,..., α r } S α F (T ). α 1,..., α r F, F (T )/F α F., E/F, K/F., EK/K [EK : K] [E : F ].. α 1,..., α n E/F, E = F (α 1,..., α n ), EK = K(E) = K(α 1,..., α n ) α 1,..., α n F, K, EK/K.. EK = K (α 1,..., α n ) E = F (α 1,..., α n ) K F 3.4.1: Lifting Property, [EK : K] [E : F ]. EK = K(α 1,..., α n ), EK α 1,..., α n K. α i 1 1 α i n n K-. α 1,..., α n E/F, α i 1 1 α i n n E α 1,..., α n F - EK α 1,..., α n K-., [EK : K] n ω ω 2 + ω + 1, F = Q, E = Q( 3 2, ω)., [E : F ] = F = Q, E = Q( 3 2, ω), K = C., ω ω 2 + ω + 1, EK = K = C, [EK : K] = 1 < 6 = [E : F ] F = Q, E = Q( 3 2, ω), K = Q(ω)., ω ω 2 + ω + 1, EK = E = Q( 3 2, ω), [EK : K] = 3 < 6 = [E : F ] distinguished class.

31 *** ( ) *** 31. (1) (Tower Property) (2) (Lifting Property) (3) (Closure under finite composites) [E : F ], [K : F ] <, , [EK : F ] = [EK : K][K : F ] [E : F ][K : F ] < F E, A E F, A. A E F (algebraic closure). F E A F, F E. F E A E.. α, β E F (α, β) α ± β, αβ, α 1 F (α, β) F., α ± β, αβ, α 1 E, A., A E. α E A, α A f(x) = a 0 + a 1 X + + a n X n A[X], F (α 1,..., α n )/F, F (α 1,..., α n, α)/f (α 1,..., α n ), F (α 1,..., α n, α)/f,, α F, α A F E, α, β E F, α ± β, αβ, α 1 F distinguished class., F {E λ } λ Λ E λ F λ Λ. (1) (Tower Property) F K E. K/F, E/K, α E f(x) = a 0 + a 1 X + + a n X n K[X] α K a 0, a 1,..., a n F, 3.4.3, M = F (a 0, a 1,..., a n ) F, M(α)/M, M(α)/F α F., E/F, K/F, E/K. (2) (Lifting Property) E/F, K/F, (2) EK = K(E) = {K(S) S E } α EK, E S = {α 1,..., α n } α K(α 1,..., α n ). α 1,..., α n E F, K, K(α 1,..., α n ) K α K. (3) (Closure under finite composites) F, E λ L, F L A. F {E λ } λ Λ, E λ A A E λ A., E λ F. λ Λ λ Λ F f(x) F [X] F., F (algebraically closed),.

32 F. (1) F. (2) f(x) F [X] F 1. (3) F [X] 1. (4) F F.. (2) (1) (1) (3) (3) (4) 3.3.3, α F, α 1. (4) (2) , f(x) F [X] f(x) = p 1 (X) p r (X)., 1 p i (X), 3.3.3, F F [X]/(p i (X))., F, F F F Ω (i) Ω (ii) Ω/F, Ω F (algebraically closure), Ω = F , C Q, C/Q. C π e Q , L/F, F L A. L, A F.. A F, A. g(x) A[X], L, g(x) = 0 α L. L α A, A L α A. A[X], A F, F Ω.. (E. Artin) F [X] {f λ (X) λ Λ}. λ λ X λ, {X λ } λ Λ F, i.e. {X λ } λ Λ R = F [..., X λ,... ]. R {X λ } λ Λ., f λ (X λ ) (λ Λ) R a., a = R 1 a N 1 = u k (..., X λ,... )f λk (X λk ) k=1., F L f λk (k = 1,..., N). L 0 = 1,., a R., a m R m. F 1 := R/m F 1 {X λ + m} λ Λ, X λ +m f λ (X λ ) = 0 F 1 /F.,, F = F 0 F 1. (1) F i+1 /F i. (2) F i [X] F i+1. Ω = F i, f(x) Ω[X], i N f(x) F i i=1., f(x) F i+1, Ω Ω.

33 *** ( ) *** 33, Ω/F. α Ω i N α F i. Tower Property F i /F α F F E. (1) E F. (2) E F. i.e., E/F, K/E K = E. (3) E F. i.e., E E F, F K E K K. 3.6 σ : F E,. (1) S F, σ S σ S. (2) a F, σ(a) a σ., S F, σ(s) C σ. (3) f(x) = a i X i F [X], σ a σ i Xi F σ [X] f σ (X). F 0 F F R σ : F R,, (embedding) F, L σ : F L σ(1) = F = L = Q( 2) σ : F L σ 1 : 2 2 σ 2 : F = Q( 2), L = Q( 3) σ : F L F = Q( 2), L = Q( 3) σ : F L σ ( 2 ) = a + b 3 (a, b Q). (.) E/F, L σ : F L F L. σ : E L σ F = σ, σ (extension)., F L, F F E F (embedding over F ),, F - (F -embedding). F L, σ : F L E, F - hom(f, L), hom σ (E, L), hom F (E, L) F = Q, E = Q( 2), L = C, σ : Q C, σ : 2 2 σ, σ hom σ (Q( 2), C) = hom Q (Q( 2), C) F = Q ( 2 ), E = Q( 4 2, i), L = C, E/F 4. σ : Q C σ ( 2 ) = 2. τ, φ : E C τ : , i i, φ : i, i i., τ ( 2 ) = 2, φ ( 2 ) = 2 φ σ, τ σ. φ hom σ (Q( 4 2, i), C), τ hom Q( 2) (Q( 4 2, i), C) (1) ( ) σ : F L f(x) F [X], α F f(x) α σ F σ f σ (X). f(x) = p(x)q(x) f σ (X) = p σ (X)q σ (X) (2) ( ) σ : K L {E λ } λ Λ K, ( λ Λ E λ)σ = λ Λ E σ λ ( λ Λ E λ)σ = λ Λ (3) ( adjoining) σ : K L F K, S K. F (S) σ = F σ (S σ ) E σ λ

34 34 (4) ( algebraic) σ : F L E/F, σ : E L σ E σ /F σ. (5) ( ) σ : F L F F, σ : E L σ E σ F σ E/F, α E F, f(x) F [X]. L, σ : F L.,. (1) β f σ (X), σ σ α σ = β. (2) σ F (α) σ (1). (3) σ F (α), f(x) F F. (, hom σ (F (α), L) cardinality α, σ L.). α f(x) n, (ii) F (α) 1, α,..., α n 1 F - γ = a 0 + a 1 α + + a n α n. σ F = σ γ σ = a σ 0 + a σ 1 β + + a σ nβ n,, σ : F (α) L. γ σ α σ = β. (2), (3) F = Q. σ : Q C. α = 2 σ E = Q( 2) σ 1 : 2 2 σ 2 : f σ (X) = f(x) = X 2 1 C F = Q( 2), E = Q( 4 2). σ : F = Q( 2) C σ : 2 2. α = 4 2 Q( 2) f(x) = X 2 2 Q( 2)[X], f σ (X) = X C[X]. L = C, f σ (X) ± 4 2i., σ E = Q( 4 2) σ 1 : Q( 4 2) C, i σ 2 : Q( 4 2) C, i E/F, L,. (1) σ : F L, σ : E L. (2), α E, f(x) F [X]. β f σ (X), (1) σ α σ = β.. E = { (K, τ) F K E, τ : K L s.t. τ F = σ α τ = β }. (K 1, τ 1 ), (K 2, τ 2 ) E (K 1, τ 1 ) (K 2, τ 2 ) K 1 K 2 τ 2 K1 = τ 1., E. {(K 1, τ 1 )} λ Λ E K = K λ, λ Λ τ : K L α K α K λ λ Λ α τ = α τ λ., τ well-defined, (K, τ) E, (K, τ). Zorn E (K, τ).,, K = E, α E \ K., 3.6.9, τ K(α) L σ : F F F F, Ω, Ω F, F, σ Ω Ω.

35 *** ( ) *** F = {f λ (X)} λ Λ F [X]. F (splitting filed), F E, (1) F f λ (X) E[X]. (2) E F F Q( 2) F = {X 2 2} Q. X 2 2 ± Q( 3 2) F = {X 3 2} Q, Q( 3 2, 3) = Q( 3 2, ω). X , 3 2ω, 3 2ω 2., ω = 1+ 3i F = {f λ (X)} λ Λ F [X].. (1) F F, F A. (2) F A 1 K 1, F A 2 K 2, A 1 K 1 F, A 2 K 2 F, F - σ : K 1 K 2 σ(a 1 ) = A 2. (3) F 2 F -.. (1) F f λ (X) ( λ Λ) 1, R, A = F (R). (2) A 1 (resp. A 2 ) F R 1 (resp. R 2 ), α R 1 α σ R 2 R 2 = R σ 1. A 1 = F (R 1 ), A 2 = F (R 2 ) A σ 1 = A 2. (3) A 1, A 2 F, A 2 L = A 2, σ := F F A 2 ( ) , σ : A 1 L., (2) σ(a 1 ) = A (2) S 1, S 2. F = Q, A 1 = Q( 3 2), A 2 = Q( 3 2ω), K 1 = K 2 = C. σ : C C, σ(a 1 ) A 2. f : A A, S A f(s) S, S f (, f- ) F K F, F F,. (1) K F [X] F = {f λ (X)} λ Λ. (2) K F - σ hom F (K, F ). (3) f(x) F [X] K f(x) K.. (1) (2) (2) (2) (3) f(x) F [X] K, f(x) β F, F - σ : K F, σ(α) = β., β σ(k) = K., f(x) 1. (3) (1) α K f α (X) K F = {f α (X)} α K F E, ( ), (normal extension), F, F E F K F, F F,. (1) K/F. (2) K F.

36 36. (1) (2) K/F, K = F (α 1,..., α n ) (α 1,..., α n ), α 1,..., α n F f 1 (X),..., f n (X) F [X], 3.7.6, f 1 (X),..., f n (X) K {f 1 (X),..., f n (X)}. (2) (1) K F {f 1 (X),..., f n (X)}, 3.7.6, K/F, f 1 (X),..., f n (X), Q ( 2 ) /Q, Q ( 4 2 ) /Q ( 2 )., Q ( 2 ) Q[X]/(X 2 2), X 2 2 Q[X] ± 2 Q ( 2 ), Q ( 4 2 ) Q ( 2 ) [X]/ ( X 2 2 ), X 2 2 Q ( 2 ) ± 4 2 Q ( 4 2 )., Q ( 4 2 ) /Q. X 4 2 Q[X] ± 4 2, ± 4 2i., Tower Property., distinguishe class., (1) F K E, F E K E. (2) (Lifting Property) F E, F E, K EK. (3) {E λ } λ Λ, F E λ, F E λ, F E λ,. λ Λ λ Λ. (1) F, K. (2) E F, F R E = F (R)., 3.1.6, EK = E(K) = F (R)(K) = F (K)(R) = K(R), EK F K. (3) σ : E λ F, σ Eλ : E λ F, σ(e λ ) = E λ. λ Λ ( ) σ E λ = σ (E λ ) = E λ λ Λ λ Λ λ Λ, 3.7.6, E λ /F., σ : E λ F σ(e λ ) = E λ λ Λ λ Λ ( ) σ E λ = σ (E λ ) = E λ λ Λ λ Λ λ Λ, 3.7.6, E λ /F. λ Λ F F, F E F., E K F K, K/F K E F (normal closure), nc(e/f ) F F, F E F. (F /F E/F.),. (1) E F, N = { K E K F F K }.

37 *** ( ) *** 37 (2) nc(e/f ) nc(e/f ) = σ hom F (E,F ). (3) α E F f α (X) F [X], nc(e/f ) F E σ F = {f α (X)} α E. (4) F S, E = F (S) α S F f α (X) F [X], nc(e/f ) F F = {f α (X)} α S. (5) E/F, nc(e/f )/F.. (1). (2) N = nc(e/f ), M = E σ. E N F, F /F, σ hom F (E, F ) σ hom F (E,F ), σ σ : F F. N/F, N σ = N, E σ N σ = N. M N., M/F. τ hom F (M, F ), σ hom F (M, F ), τσ hom F (M, F ) τ(m) = τ σ(e) = τ (σ(e)) = M σ hom F (E,F ) σ hom F (E,F ), M/E. F M F, N N M. (3)(4). (5) N = nc(e/f ). E/F, 3.4.3, E F, α 1,..., α n. f 1 (X),..., f n (X) F [X] N, (4), 3.4.3, N/F F, f(x), g(x) F [X]. K f(x), g(x). ( K F ),. (1) f(x), g(x) F [X] monic d(x) F., d(x) K[X]. (2) a(x), b(x) K[X] a(x)f(x) + b(x)g(x) = d(x).. K[X] PID, K[X] (f(x), g(x)) K = (d 0 (X)) K monic d 0 (X) K[X],, a(x), b(x) K[X] a(x)f(x) + b(x)g(x) = d 0 (X)

38 38. K F d 0 (X) (f(x), g(x)) F, f(x), g(x) F [X] monic d 1 (X), d 0 (X) (f(x), g(x)) F = (d 1 (X)) F d 1 (X) F d 0 (X)., f(x), g(x) (d 0 (X)) K d 0 (X) K f(x) d 0 (X) K g(x) d 0 (X) F [X] f(x), g(x), d 0 (X) f(x), g(x) d 0 (X) F d 1 (X). monic d 0 (X) = d 1 (X) F, f(x) F [X]. 2 f(x) F [X] (separable) F f(x)., (inspeparable) F, f(x) F [X].,. (1) f(x) (2) f(x) f (X) F [X] (3) f (X) 0. f(x) F [X] f(x) = (X α 1 ) e1 (X α r ) e r F [X], F [X] (1) (2), F [X] (1) (2). (2) (3), (2) (3). f(x), f(x) f (X) F [X] d(x) 1 f(x). deg f (X) < deg f(x) d(x) f(x) f (X) 0. d(x) = 1, f(x) f (X) ch F = 0 F f(x) F [X]., subsection ch(f ) = p F ch(f ) = p 0, f(x) F [X]., f(x) f(x) = g (X pd) d > 0 g(x) F [X]., d g(x), d f(x) (radical exponent), f(x) p d.. f(x) = a 0 + a 1 X + + a m X m f (X) = 0 p i a i = 0 f(x) = a 0 + a p X p + a 2p X 2p + + a pl X pl = q(x p )., q(x) = a 0 + a p X + a 2p X a pl X l., q(x),., f(x) = g(x pd ) g (X) 0. g(x) g(x) = (X α 1 ) (X α k ), g (X) 0. f(x) f(x) = (X pd α 1 ) (X pd α k ) = (X pd β pd 1 ) (Xpd β pd k ) = (X β 1) pd (X β k ) pd, p d F. 2,

39 *** ( ) *** 39. F, p > 0, F p, n F q = p n. F q 1, F α 0 α q 1 = 1. α α q = α, α F. f(x), f(x) = g(x p ) = a 0 + a 1 X p + + a m X pm. a i = b p i (i = 1,..., m), f(x). f(x) = b p 0 + bp 1 Xp + + b p mx pm = (b 0 + b 1 X + + b m X m ) p 3.11 S F, n, { s n s S } S n ch(f ) = p 0, E/F, S E. (1) F (S) = F (S pk ) 1 k 1, k 1. (2) F = F pk 1 k 1, k 1.. (1) 1 k 0 1 F (S) = F (S pk 0 )., F (S) = F (S pk 0 ) F (S p ) F (S) F (S) = F (S p ). k 1, S pk F (S) F (S pk ) F (S). k. k = 1. k, F (S) F (S pk ). α F (S) = F (S p ), S p S p F -, S p s pi 1 1 s pi k k., s 1,..., s k S F (S pk ), S pk. s j S pk. (2) F = F pk 0 F -, F (S pk+1 ) F - 1 k 0 1, F = F pk 0 F p F F p = F., k 1 F pk = (F p ) pk = F pk+1., (3) E/F, L σ : F L., hom σ (E, L) E/F, L σ., L τ : F L, hom σ (E, L) = hom τ (E, L ).

40 40 L λ = τσ 1 L σ E τ σ(f ) σ F τ τ(f ) : E/M/F. E/F, σ(f ), τ(f ),, L, L σ(f ), τ(f )., τσ 1 : F σ F τ λ = τσ 1 : L L , λ(l) = L λ. σ hom σ (E, L), λσ : E L hom τ (E, L ),, τ hom τ (E, L ), λ 1 τ : E L hom σ (E, L), hom σ (E, L) = hom τ (E, L ) E/F. L, σ : F L hom σ (E, L) E F (separable degree), [E : F ] s α E F, α F f(x) F [X]. f(x), α (separable)., f(x), α (inseparable), f(x) α (radical exponent) E/F, α E F f(x) F [X]. L σ : F L.,. (1) α [F (α) : F ] s = [F (α) : F ] (2) α, α d hom σ (F (α), L) [F (α) : F ].. [F (α) : F ] = deg f F K E, [F (α) : F ] s = 1 [F (α) : F ] pd [E : F ] s = [E : K] s [K : F ] s.. σ : F E. σ K σ hom σ (K, E) [K : F ] s, σ τ : E E σ [E : K] s, [E : F ] s [E : K] s [K : F ] s., τ hom σ (E, E), τ K : K E hom σ (K, E), τ τ K, [E : F ] s [E : K] s [K : F ] s., E/F (separable) α E F., (inseparable) ( ) E/F, ch(f ) = p 0., 4.

41 *** ( ) *** 41 (1) α F. (2) [F (α) : F ] s = [F (α) : F ] (3) F (α)/f. (4) k 1 F (α) = F (α pk ). (, , k 1.), α F α F d (1) (2). [F (α) : F ] s = 1 [F (α) : F ] pd (1) (3). β F (α) F F (β) F (α) , [F (α) : F ] s = [F (α) : F (β)] s [F (β) : F ] s, , [F (α) : F ] = [F (α) : F (β)][f (β) : F ], (2) [F (α) : F ] s = [F (α) : F ], , [F (β) : F ] s [F (β) : F ] [F (β) : F ] s = [F (β) : F ]., β F. (3) (1). (1) (4). F K F (α), α F f(x), K g(x), f(x) K[X], f(α) = 0, g(x) f(x). k 1 F F (α pk ) F (α), α F (α pk )[X] X pk α pk = (X α) pk, α F (α pk ) f(x) (X α) pk, α F (α pk ) F (α pk ) = F (α).. α F (α pk ), f(x) = X α (4) (1) , k 1, F (α pk ) = F (α)., d α, α pd F, (1) (3), F (α pd ) = F (α), α. α F f(x), f(x) = g ( X pd), g(x) (3) [F (α) : F ] s f(x) [F (α) : F ] s = deg g. f(x). [F (α) : F ] = deg f = p d deg g = p d [F (α) : F ] s, ( ) E/F, ch(f ) = p 0., 4. 1) E/F. 2) [E : F ] s = [E : F ] 3) F α 1,..., α n E E = F (α 1,..., α n ). 4) S E E = F (S), k 1 E = F (S pk ). (, , k 1.), E/F [E : F ] s = 1 p e [E : F ] e 1.

42 42. 1) 3) E/F, 3.4.3, S S 0 = {α 1,..., α n } E = F (S 0 ). E/F, α 1,..., α n F. 3) 2) 3.4.3, F α 1,..., α n E E = F (α 1,..., α n ). F F (α 1 ) F (α 1, α 2 ) F (α 1,..., α n ) = E, α i F (α 1,..., α i 1 )., α i F. 3, , [F (α 1,..., α i ) : F (α 1,..., α i 1 )] s = [F (α 1,..., α i ) : F (α 1,..., α i 1 )] i = 1,..., n , [E : F ] s = [E : F ]. 2) 1) β E, F F (β) E, , [E : F ] = [E : F (β)][f (β) : F ], [E : F ] s = [E : F (β)] s [F (β) : F ] s,, [F (β) : F ] s < [F (β) : F ] [E : F ] s = [E : F ]. [F (β) : F ] s = [F (β) : F ], β F. 1) 4) E S, E = F (S). α S F, F (α) = F (α pd ) F (S pk ) k 1. F (S) = F (α 1,..., α n ) F (S pk ). F (S pk ) F (S). 4) 1), F (S pk ) = F (S) k, , k 1., k S d α S α pk. 4 F (S pk ) F ( ) E/F, ch(f ) = p 0.,. (1) E/F S E = F (S). (2) E/F E = F (S), k 1 E = F (S pk ).. (1). E/F E F., F S E = F (S). β E, (2), S S β F (S 0 )., 3.4.3, F (S 0 )/F, ,., β F. E/F. (2). α S k (4) F (α) = F (α pk ) F (S pk ), F (S) F (S pk )., F (S) = F (S pk ). 3 F K E, α E F, α F f(x). α K g(x) f(x) K[X] f(α) = 0, g(x) f(x). g(x). 4 β F, k 1 β pk F., β F, (3) F (β)/f, β pk F (β) F.

43 *** ( ) *** distinguished ( distinguished) (1) distinguished class. (2) (composite). (3) E/F, nc(e/f ) F.. (1) Tower property,, F K E, E/F, K/F. E/K, α E F f(x) F [X], K g(x) K[X] g(x) f(x), f(x) g(x)., K/F, E/K, α E K g(x) K[X] g(x) S g(x), F (S, α)/f (S)., F (S)/F F (S, α)/f (S), F (S)/F, , , [F (S, α) : F ] s = [F (S, α) : F (S)] s [F (S) : F ] s = [F (S, α) : F (S)][F (S) : F ] = [F (S, α) : F ], α F. Lifting property, E/F K/F, α E F, K., EK = K(E) K. (2) E λ /F, E λ E λ, ). λ λ (3) nc(e/f ) E F.,, nc(e/f ) F, ) F, F (perfect) F F.. ( ) F α E F, f(x) F [X]., , E/F. ( ) f(x) F [X], f(x) α F (α)/f., f(x) ,.., F ch(f ) = p 0,. (1) F. (2) k 1 F = F pk. (3) k 1, (Frobenius map) σ p k : x x pk F., 2) 3), k 1.

44 44. 1) 2) F, α F, f(x) = X p α F [X]. {f(x)} E, β E f(x), β p = α f(x) = X p β p = (X β) p. β F g(x) g(x) f(x), g(x) g(x) = X β., β F., α F β F, β p = α. F F p. F p F F = F p. 2) 3) F = F pk σ p k : x x pk.. 3) 1) σ p k : x x pk F = F pk, (2), m F = F pm., f(x) F [X], f(x) = g(x qd ) g(x). g(x) = i a ix i b i F b pd i = a i, f(x) = i b pd i (X qd ) i = ( i ) p d b i X i f(x) ) F E/F E. 2) E E/F F.. 1) ) ch(f ) = p 0. E F. E = F (α), α F. F (α) , F (α) = (F (α)) p = F p (α p ). 5 α f(x) = a i X i F [X] i ( ) p 0 = a i α i = i i a p i αpi [F p (α p ) : F p ] [F (α) : F ]. F p F F (α) = F p (α p ), [F p (α p ) : F p ] = [F (α) : F ][F : F p ] [F : F p ] = 1, F = F p. E/F, 3.4.3, E F, E p = E , E F α (purely inseparable), α F α 1. E/F, E F, E/F (purely inseparable extension)., α F F, α F f(x) = (X α) n ( ). X n 1 nα, n p = ch(f ). ( n 0 nα F α F ) n p k f(x) = (X α) mpk 5 F (α) F - α a i α i, p (F (α)) p i α p F p (α p ). ( ) p a i α i = i i a p i αpi F p (α p ), F p -

45 *** ( ) *** 45 (m p ) , g(x) f(x) = g(x pd ). f(x) mp k, g(x) r rp d = mp k k d r = mp k d. f(x) α f(x) = (X α) mpk = {(X α) pd } mpk d = (X pd α pd ) mpk d g(x) = (X α pd ) mpk d, α, g(x), m = 1, k d = 0. f(x) = (X α) pd = X pd α pd, d α ch(f ) = 2 t F, t F (t 2 ), X 2 t ,,. ch(f ) = p 0, α F 0, t F., tp2 s = t p + α, F (s) F (t), p 2. t monic f(x) = X p2 sx p sα ), p 2,. f(x) = g(x p ), f(x).,, t F (s). f(x) = X p2 sx p sα = (X t) p2 = X p2 t p2, s = 0,. t F (s), E/F, , [E : F ] s [E : F ] [E : F ] = [E : F ] s [E : F ] i. [E : F ] i E F (inseparable degree)., F E ch(f ) = p 0. 1) F K E [E : F ] i = [E : K] i [K : F ] i. 2) E/F [E : F ] i = 1. 3) α E d [F (α) : F ] i = p d. 4) α E [F (α) : F ] s = 1, [F (α) : F ] i = [F (α) : F ]. 5) [F (α) : F ] i p.. 1) , ) ) (2) 4) α E α F f(x) 1. hom F (F (α), F ) = 1. 5) ch(f ) = p 0, α F, α d, f(x)., 3. 1) α E.

46 46 2) F (α)/f. 3) k 0 α pk F.,, d α pk F.. 1), β F (α), F F (β) F (α), ) [F (α) : F ] i = [F (α) : F ]., , ) [F (β) : F ] i = [F (β) : F ]., ) β. 2) 1) trivial. 1), f(x) = X pd α pd, α pd F., 3), g(x) = X pd α pd, g(α) = 0 f(x) g(x), f(x) α, f(x) ) F (α pd ) = F., α, k F (α pk ) = F (α). ( , (4) ) E/F, 3. 1) E. (, E F (purely inseparably generated).) 2) [E : F ] s = 1 (, E F (degreewise purely inseparable).) 3) E/F.. 1) 2) E = F (I), I F. L, σ : K L I., α I, f(x), α σ f(x). f(x) α α σ = α σ. [E : F ] s = 1. 2) 3) α E, α α, β F f(x), ι : F F α σ = β σ : E F. [E : F ] s = 1 σ E, β = α f(x) 1, α F. 3) 1) distinguished class.,.. F K E ,, [E : F ] s = 1 [E : K] s = 1 [K : F ] s = 1, tower property. Lifting property. E/F K/F., E F, K. EK = K(E) K E, ) EK/K.. E λ /F E λ F, E λ F E λ, ) F F q n monic. 1 n d n ( n ) µ q d d., F 2 2 X 2 + X X 3 + X + 1, X 3 + X 2 + 1

47 *** ( ) *** : F q n q = q = q = q = X 4 + X + 1, X 4 + X 3 + 1, X 4 + X 3 + X 2 + X X 5 + X 2 + 1, X 5 + X 3 + X 2 + X + 1, X 5 + X 4 + X 3 + X X 5 + X 3 + 1, X 5 + X 4 + X 2 + X + 1, X 5 + X 4 + X 3 + X X 6 + X + 1, X 6 + X 3 + 1, X 6 + X X 6 + X 4 + X 2 + X + 1, X 6 + X 5 + X 2 + X + 1, X 6 + X 4 + X 3 + X + 1 X 6 + X 5 + X 4 + X + 1, X 6 + X 5 + X 3 + X 2 + 1, X 6 + X 5 + X 4 + X , F 3 2 X 2 + 1, X 2 + X 1, X 2 X 1 3 X 3 X + 1, X 3 X 1, X 3 + X 2 1, X 3 + X 2 + X 1, X 3 + X 2 X + 1, X 3 X 2 + 1, X 3 X 2 + X + 1, X 3 X 2 X 1 4 X 4 + X 1, X 4 X 1, X 4 + X 2 1, X 4 X 2 1, X 4 + X 3 1, X 4 X 3 1, X 4 + X 2 + X + 1, X 4 + X 2 X + 1, X 4 + X 3 X + 1, X 4 X 3 + X + 1, X 4 + X 3 + X 2 + 1, X 4 X 3 + X 2 + 1, X 4 + X 3 + X 2 + X + 1, X 4 + X 3 + X 2 X 1, X 4 + X 3 X 2 X 1, X 4 X 3 + X 2 + X 1, X 4 X 3 X 2 + X 1, X 4 X 3 + X 2 X + 1.

48

49 4 4.1,,,,,,., ,,,,. 1828, 17,,, Louis-le-Grand,., ,.,., ,.,,.,.,, (the Grand Prize in Mathematics of the Academy of Sciences).,, ,, ,,. 5 14, , 1831,,,., ,,.,,,,,,,,.,. 1 1 Evariste Galois life was, to say the least, very short and very controversial. Of course, it would not be the subject of such legend today were it not for his remarkable discoveries, which spanned only a few short years. Galois was born on October 25, 1811, near Paris. Apparently, Galois was recognized at an early age as a brilliant student with some bizarre and rebellious tendencies. In 1828, at the age of 17, Galois attempted to enter the prestigious Ecole Polytechnique, but failed the entrance exams, so he remained at the royal school of Louis-le-Grand, where he studied advanced mathematics. His teacher urged Galois to publish his first paper, which appeared on April I, After this, things started to go very badly for Galois. All article that Galois sent to the Academy of Sciences was given to Cauchy, who lost it. (Apparently, Cauchy had a tendency to lose papers; he had already lost a paper by Abel.) On April 2, 1829, Galois father committed suicide. Galois once again tried to enter the Ecole Polytechnique, but again failed under some rather controversial circumstances. So he entered the Ecole Normale, considered to be on a much lower level than the Ecole Polytechnique. While at the Ecole Normale, Galois wrote up his research and entered it for the Grand Prize in Mathematics of the Academy of Sciences. The work was given to Fourier for consideration, who took it home, but promptly died, and the manuscript appears now to be lost. Galois possessed very strong political opinions. On July 14, 1831, he was arrested during a political demonstration, and condemned to six months in prison. In May 1832, Galois had a brief love affair with a young woman. He broke off the affair on May 14, and this appears to be the cause of a subsequent duel that proved fatal to Galois. Galois died on May 31,1832. On September 4,1843, Liouville announced to the Academy of Sciences that he had discovered, in the papers of Galois, the theorem, from his 1831 Memoir, that we mentioned earlier concerning the solvability by radicals of a prime degree equation, and referred to it with the words as precise as it is deep. However, he waited until 1846 to publish Galois work. In the 1850s, the complete texts of Galois work became available to mathematicians, and it initiated a great deal of subsequent work by the likes of Betti, Kronecker, Dedekind, Cayley, Hemiite, Jordan and others. Now it is time that we left the past, and pursued Galois theory from a modern perspective. 49

50 P Q. π : P Q ω : Q P (π, ω), 2 (Galois connection). 1) ( (order-reversing antitone)) p, q P, r, s Q p q p π q π r s r ω s ω 2) ( extensive) p P, r Q p p πω r r ωπ P. cl : P P, 3 (closure operation)., p, q P. 1) ( extensive) p cl(p) 2) (idempotemt) cl(cl(p)) = cl(p) 3) (Isotone) p q = cl(p) cl(q) p P cl(p) = p p (closed), P Cl(P ) (π, ω) (P, Q). p p πω q q ωπ, P, Q, p πω = cl(p), q ωπ = cl(q). 1) p πωπ = p π, cl(p π ) = cl(p) π = p π 2) q ωπω = q ω, cl(q ω ) = cl(q) ω = q ω.. p p πω π p π (p πω ) π = p πωπ = (p π ) ωπ p π, p π = p πωπ., q ω = q ωπω, π : P Cl(Q) ω : Q Cl(P ), π : Cl(P ) Cl(Q) ω : Cl(Q) Cl(P ) (order-reversing bijection) P, Q, (π, ω) (P, Q). 1) P Cl(P ). Q. 2) P, Q (De Morgan s, Law). p, q P, r, s Q (p q) π = p π q π (p q) π = p π q π (r s) ω = r ω s ω (r s) ω = r ω s ω.. 1). 2) p λ = max{x P x p λ for λ}, p λ = min{x P x p λ for λ} λ λ p q = max{x P x p x q}, p q = min{x P x p x q}..

51 *** ( ) *** 51 cl, P 1 P. (,,.) 1 P cl(1 P ) 1 P cl(1 P ) = 1 P., 0 Q 1 P = 0 ω Q., 1 P = cl(1 P ) = 1 πω P 1 πω P = 1 P 1 P = 0 ω Q. 1π P 0 Q 1 πω P 0ω Q 1 P,,., 1 P Q closed 1 π P.2 Q closed 0 Q = 1 π P X Y, P = P(X) Q = P(Y ) P Q. R X Y X Y. 3 S P(X) S π = {y Y (x, y) R for x S} P(Y ) T P(Y ) T ω = {x X (x, y) R for y T } P(X) (π, ω) (P(X), P(Y )) (P, Q) (π, ω) (indexed) a) p q p, q P, q p (degeee index) (q : p) P Z >0 b) r s r, s Q, s r (degeee index) (s : r) Q Z >0, 3., P, Q (q : p). 1) (Degree is multiplicative) s 1, s 2, s 3 P s 1, s 2, s 3 Q s 1 s 2 s 3 = (s 3 : s 1 ) = (s 3 : s 2 )(s 2 : s 1 ) 2) (π and ω are degree-non-increasing) p, q P p q = (p π : q π ) (q : p) r, s P r s = (r ω : s ω ) (s : r) 3) (Equality by degree) s, t P s, t Q s t (s : t) = 1 s = t (s : t) < s t (finite extension). P, index(p ) = (1 P : 0 P ) P (index). Q., (q : p), p q (π, ω) (P, Q).,. 1) (Degree-preserving on closed elements) p, q Cl(P ) p q (q : p) = (p π : q π ). Q. 2) (Finite extensions of closed elements are closed) p Cl(P ) (q : p) < q Cl(P )., 0 P (1 P : 0 P ) P. Q.. 1) (q : p) (p π : q π ) (q πω : p πω ) = (cl(q) : cl(p)) = (q : p). 2) p Cl(P ) (q : p) < (q : p) (p π : q π ) (q πω : p πω ) = (cl(q) : p) = (cl(q) : q)(q : p) (cl(q) : q) = 1 q. 2 1 πωπ P = 1 π P 1π P. q Q qω 1 P q = q ωπ 1 π P. 3 A B A B R.

52 (π, ω) (P, Q). p, q P, 1) (q : cl(p)) < (q : p) = (p π : q π ) 2) cl(p) q (q : p) = (p π : q π ) < p., q = 1 P, (1 P : p) = (p π : 0 Q ) < p.. 1) (q : cl(p)) <, ) q closed, ) cl(p) π = p π, ) > (q : cl(p)) = (p π : q π )., (q : p) = (p π : q π ) (p π : q π ) = (q : p) = (q : cl(p))(cl(p) : p) = (p π : q π )(cl(p) : p), (cl(p) : p) = 1 p = cl(p). p closed., 2) (q : cl(p)) (q : p) < 1)., (P, Q) (π, ω), P, P. Q. P Q, (P, Q). 1 P, 1 Q, closed. 0 P, 0 Q., ( ) 0 Q closed,, (0 Q is closed) (π, ω) (P, Q). P, Q. 0 Q index(q) index(p ). 1) index(q) < index(p ) < Q. 2) index(p ) < 0 P, (P, Q).. index(p ) = (1 P : 0 P ) (0 π P : 1 π P ) = (1 Q : 1 π P ) = (1 Q : 0 Q ) = index(q) P Q index(q), Q. 2), (Galois Correspondence) E/F, E F Aut F (E) G F (E), E F (Galois group of E over F ). E/F, E/F G F (E) = Aut F (E) = hom F (E, E) G F (E) = hom F (E, E).