Теория функциональных полей над конечным полем констант

2 ..........................................................................................................................................3 1. 1.1. 1.2. 1.3. p1.4. 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 3. 3.1. 3.2. 3.3. 3.4. P3.5. 4. 4.1. 4.2. AG4.3. AG4.4. 4.5. 4.6. 5. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 6. 7. 7.1. 7.2. 8. 8.1. 8.2. 8.3. .............................................................................................4 .............................................................................................................................4 .....................................................................................7 * ..............................................................9 ............................................................................10 ....................................................................................................12 ..............................................................................................................12 ........................................................................................................................................12 ..................................................................................14 .........................................................................................15 ..................................................................................................................................16 . ..........................................................................18 ...............................................................20 ................................................................................................................20 .......................................................................................................................21 – ............................................................................................................23 ..........................................................................................................24 ....................................................................................................................................26 ......................................................................................28 *.............................................................................................................28 CL (D, G) ...................................................................................................................28 C (D, G)...................................................................................................................30 AG.....................................................................................................31 AG...........................................................................................................33 AG.........................................................................................................36 ......................................................................40 .................................................................................................................................40 .....................................................................................................................42 ............................................................................................................................44 .....................................................................................................46 ...............................................................................................................47 – ..............................................................................................48 * .....................................................................................50 .......................................................................................................................53 ..................................................................55 ..............................................................................................55 ......................................................................58 *.................................................................................................61 *......................................................................................................61 * ...........................................................................................................................61 , , *..............................................................................63 ............................................................................69 3 . , , . . , - . . - , . XIX . , . XX , . . , , , , - . , , . , . , , , [4]. [5] . [6] . . , , , - [7] [3]. [1] , [2]. , . , , , , . , - , . – , . - , . 4 1. 1.1. 1.1.1. . , K/k z K , K/k - k. 1.1.2. K/k ( ). y 1.1.3. . - K( ), k , K = k(y). , - : 1) char k = 0, 2) char k = p > 0 y p p k = {x | x k; k} = k ( . x k p , :x x p - k - k). 1.1.4. . 1.1.5. . , . . K= k ) , ( 1.1.6. K= k, . k L K. K L L. x 1.1.7. . k, K, xp n L K (x) = x GL. k– p > 0, K – k, . x K K x n 0, k a k - k. 2) . k. 1.1.8. : 1) n , k, : k x , - k. L– . L– GL = Aut (K / L) , . q x Tp K n a k. 3) S K, , K = k(S), s S - k. 1.1.9. , . 1.1.10. K, 1) K 2) . K / k, . K/k– k. k K). - K0 – K, , K0 – k ( K0. K/k– , K0 / k – . - 5 3) K / k– G K, , G– G K , K G k K K – k. G 1.2. 1.2.1. . A , A– , A\A A, ( - A\A . , p = A\A A– ). . /8 1.2.2. x . n p = 2 /8 . . A/R– 1 . a0, a1,..., an n x + an 1 x x A, R, 1 n 1 x R + ... + a1 x + a0 = 0. A, A R. R ( - R=R, A. . 1.2.3. R, - , , R, ), A R; R - , ). K/k– , K[T1,…, Tn] / k[T1,…, Tn] – . 1.2.4. K, . x K , K– x , 1.2.5. 1) A – 2) A . A, x x x 2) K - . A. A– K. . K. x – y A\A . y = x(1 a) A \ A , y = y(a , A– A A. K 1) , 1 1 1) x A\A n . . K, x– . y x, y 0. , A. a0,…, an 1 1 n x + an 1 x x n 1 z x A, , A, , A, (n 1) , 1 n 1 = (x ) A, x + an 1 + … + a1 x x A. y = 0, a A, . a xz A\A . . x 1 1 A, A. + … + a1 x + a0 = 0. , x , A\A . x=0 a = y / x K. x n+2 + a0 x . n +1 = 0, a0 0. - 6 1.2.6. : a) A – . A, , - p b) t A, . , z n z = ut , n– 0, u 1) (1.1) A . 2). p. t– (1.1) : ut n m =v , u = v. . A, : , n t, z– , , z = z1 t z1 = z2 t. , , A , (zm) = (zm +1) = … , zm = y t zm y t = 1. n 1 z n=m A z1 A. z1 – , t– zi = zi +1 t (z1) (z2) … m. zm +1 = y zm , t– u– . zn 1 = u t, 2 1) : n 1 , z = ut n 1. , x = yt y A– . p = (t), , - a– - n n (u – a x n zi n . a (1.1), - = ut . , , A. m. . z = z1 t = z2 t = … = zn 1 t , u, v – z1, z2,... , , 2) m ut = vt , , : y n , (1.1) (1.1) : , , . , A m x = vt , v– m ). n. - , a = (t ) = (t) = p – n A. n n (t) – . . . 1.2.7. , . , . t, A. , = A / p, , 1.2.8. 1) A. . A– t– n z = ut , 2) A 3) A A, n z , u . n. 1) 2) 3) , K– . 1). 1.2.6. . K - A . K. a A a=t A n - 7 1.2.9. . A– . A, t A p t– A. 1.3. 1.3.1. K 1) 2) 3) .( , v: K x, y K : x, y K : x K: ) : v(xy) = v(x) + v(y), v(x + y) min(v(x), v(y)), v(x) = x = 0, , v1 c > 0, ( a a< , a+ 1.3.2. a) - { }, : = , + = . v2 K v1 = cv2. , , ( v(K ) ) K - ): v , - v. b) c) x, y K v(x) n x K, n v(y), v(x ) = n v(x). v(x + y) = min(v(x), v(y)). 1.3.3. . K | |:K , : 1) 2) 3) x K: |x| 0 |x| = 0 x = 0, x, y K : | xy | = | x || y |, x, y K : | x + y | | x | + | y |. | | 3) x, y K: |x + y | ( , max{| x |, | y |}. ( ( 3) x, y K: |x| 1.3.4. x, y , K– |y | ) | x + y | = max{| x |, | y |}. . 1) , , | |. K , - K v– | 0 | = 0. , c1 , c2 – :) . K, c > 1. |x| = c a) b) ( d(x, y) = | x – y |. . , 1.3.5. ) , K ) x K, x 0, v(x) . K . > 1, | |1 . | |2 - 8 c) , - . 2) , | | K, c > 1. x K, x - v(x) = log c | x |. , v(0) = . a) b) , K c1 , c2 – v2 , . v1 c) . - 1.3.6. a) v b) v ). = {x = {x v 2) v v– . 0} = {x K | x | K v(x) K v(x) = 0} = {x c) pv = {x 1) v. > 1, , K v(x) > 0} = {x K, | | 1}, K | x | = 1}, K | x | < 1}. – K ( , K– v. 3) pv = v v \ v. v v / pv = v. 1.3.7. . v K , s. v(K ) = s v t K, , , v(t) = s = 1. - t v– K. t v 1.3.9. , v v, . v(t) = 1. v– K, | | – c > 1, p = pv – , K, | |, p =t n n n z = yt , v y n n (p )n v. p- . v(z) = v(y) + nv(t) v , p – (p )n ). s = 1. v. ). 2) n - ( , 1.3.8. K. 1) v – z v). n 1/c . z n K |z| n 1/c . , - 0. x (x + p )n n K (x + p ) x n K, n p x. K. 9 U K , x U x+p n . 1.4. p- * 1.4.1. p– . x = a/b x = pn n– ( p 1.4.2. . , vp : { } , vp (x) = n. , - . . p. x , x 0, | x |p = p 1.4.4. , - a1 , b1 p 1.4.3. a1b1. vp (x) vp (0) = . ), x = a / b. , x 1.3.5 1) v p ( x) | 0 | p = 0. , | |p : K - p, . p- 1.4.5. 1) . . K= | | = | |p p – = (p) p- . = {a / b p b} . , (p) = p , p. 2) pp = p p. p = {a / b p b, p a}. – p. 3) p p = {0, 1,…, n – 1} – 1), 2) . a/b n n , |a/b| = p a/b n > 0, p, . , a/b p a1 b1, p 0. p / pp p a / b = p a1 / b1, a/b = , , p a, p b. 0, n n = vp (a / b), n 1, , |a/b| = p . , , p b. , . a/b pp 10 3). , p = (p). , 0, p / pp. 1,…, p – 1 ) m, n , ) n (mod pp). {0, 1,..., p – 1} m p m–n pp. , m = n, a/b p, s1 n 1 , s p + nb = a, , . . p b. , n (mod pp). n , a/b r (mod pp). . m – n = pa / b, b . - s = a s1, n = a n1. p. p b, 0, 1,..., p – 1 , p s1 p + n1 b = 1. pp, m–n a a nb ps n = = b b b . a/b , a pp, n = p q + r, r < p. r (mod pp). n pp , 0, 1,..., p – 1. , {0, 1,..., p – 1} , ( p p / pp = = p - pp , – p 1.4.6. . . ). ) | | p, , p– | |. 1.5. 1.5.1. k, k(x) – z k(x) g(x) 0. 1.5.2. k– , k[x] – x ( ) : z = f (x) / g(x), p(x) – k[x]. z = p ( x) n n– ( 1.5.3. p(x) ), z = f (x) / g(x). . vp(x) : k(x) , x k. f (x), g(x) – z = f (x) / g(x) k[x] k(x), f (x)g(x) - f1 ( x ) , g1 ( x) f1 (x)g1 (x). vp(x) (z) vp(x) (0) = . vp(x) (z) = n. , . { } k(x) . p(x) . 1.3.5 1) 1.5.4. , 1) k(x) . 1.5.2. k(x) – . , vp(x) – vp(x) p(x) = k [x](p(x)) = {f (x) / g(x) k(x) p(x) g(x)} - 11 – k [x] (p(x)) = p(x)k[x] k [x] k [x]. , , - p(x). 2) vp(x) Pp(x) = p(x) p(x). p(x) = {f(x) / g(x) k(x) p(x) g(x), p(x) f(x)}. – p(x). 3) Fp(x) = p(x) = x , x p(x) / Pp(x) , Px k[x] / (p(x)). , P Fx F 1.4.5 ( 1.5.5. z = f (x) / g(x) k(x), f (x)g(x) . ). v (z) = deg g(x) – deg f (x). ( ), z = f (x) / g(x). . 1.5.6. . , v : k(x) 1.5.7. . 1.5.5. 1 / x. k(x) – 1) , { } . k(x) , v (z) v (0) = . , v – - v = {f(x) / g(x) , , k [1 / x] 2) k(x) g(x) 0, deg f(x) deg g(x)}. . v P = p(x) 1 / x. = {f(x) / g(x) k(x) deg f(x) < deg g(x)}. – . 3) F = /P k. 1.4.5 ( 1.5.8. v vp(x), . k(x) p(x) – ). , k [x]. 12 2. 2.1. 2.1.1. F / k, . F k K k(x). ) ( k(T) x F, , F k F . k(x), x, k 2.1.2. F/k - T . , . . k k k F / k(x) x . - F/k , F, x F / k. 2.1.3. . k F/k– x . F , 2.1.4. , x char k = 0, char k = p > 0, . p F . x F/k – F k. y F, , F = k(x)(y) = k(x, y). y, , k(x)[T] g k(x). T, , y, . g(y) = 0. (2.1) (2.1) k[x], f (x, y) = 0, , . , f (2.2) k, . F/k – v(x) = 0 (2.2) k[X, Y], X, Y – . - 2.2. 2.2.1. . x , k, x , v 2.2.2. k, . v– F / k. P = pv 1.3.6 – 1.3.7 F / k. F / k, ) . P. ( ) v P - v t v F k. v ( v 0. = {x F x P 1 P}, P. 13 , 2.2.3. . . – F / k, k. 1) – F. 2) P– , P n = (0). (2.3) n 1 2.2.4. , P vP P. t =P 1) F / k, (2.3) n z - P =t n P z . z= t 1 . … , z P n …. 0, n n . z P n +1 , 0, P , , , – n 0. . 2) z z= , 1 1 z n , 1 t , , z – , , F z 1 = t n n– . n z= t , , n . 3) 4) , 1.2.6 vP (z) = n. , . , vP (0) = . 1.4.2 vP – , 5) vP vP = {x F vP (x) , P– 2.2.5. | |P – | |P, 2.2.6. , pvP = {x 0} = F / k, k. ( ) F vP (x) > 0} = P. vP. P– F / k. . P- - vP F, - . . F / k, - k, F / k. F/k 2.2.7. . P k– F, F. z t . p > 0, z F / k. P F vP (z) 0 (mod p) , - 14 2.2.8. . 1.5.8 Pp(x), P , k(x) k[x]. p(x) – 2.3. 2.3.1. . 1) FP = P/ P F / k. P P - k. 2) [FP : k] , , k= , q 3) d = deg P FP = qd ). deg P = 1 2.3.2. . Pp(x), . 1.5.4 p(x) – 1.5.7 , k[x] / (p(x)), , deg Pp(x) = deg p(x). , P k, k. , P . , k= k(x) k[x] Fp(x) P P ( - k(x) , q 2.3.3. q+1 . . z P P, k. . P– F / k, P : P FP - F P ( z ), z(P) = F P, F\ P. z , FP z { }, z z(P) P. 2.3.4. ). k(x) z(P ) = z= 0, g( ) 0. , nx n ... mx m ... z(P ) = 0, , 1) P– , z = f(x) / g(x) f ( ) / g ( ), g ( ) n 2.3.5. / n, k(x), n m, n m, n m. F / k, x, y, z P, k. , (x + y)(P) = x(P) + y (P), - (xy)(P) = x(P) y (P), (P) = . k, 15 2) z(P) = 0 z. 3) z(P) = z P z n = vP (z) > 0 vP (z) > 0. , vP (z) < 0. P P , P z. 2.3.6. . k z F , 2.3.7. . : k , FP P FP , , . FP. P n = vP (z) P K K. K P. K, P :K FP - FP , FP P FP . 2.3.8. , 2.3.7 z – z(P) Ker P z P (z) P = P. , P k P = FP = FP + P. 2.3.9. ( P (z(P)), FP P = {0}, (2.4) , =k P– 1, P. (2.5) . , z k ), F z(P) z: k . z F/k FP = k P. { }. k F - P ). P = z(P) = , { }, P z - z(P), . , F/k - . 2.4. 2.4.1. ( ). F / k, z1,…, zn F, r1,…, rn vPi ( z zi ) = ri 2.4.2. a1,…, an k. P1,…, Pn – 2.4.1 j ij – . FPj Pj F. , 1 , i z F, , n. F/k zj {1,..., n} vPi ( z j ) = 1 – P1,…, Pn – k, F, , i j, zj (Pi ) = 0 i j zj (Pj) = cj – - 16 n a j c j 1z j z= F. (2.6) j 1 i {1,..., n} z(Pi ) = ai. P1,…, Pn , k zj (Pj) = cj k (2.6) z . . 2.4.3. ( ). 2.5. 2.5.1. . F/k D= nP P P P vP (D) = nP, nP = 0, , , - . 1) P, nP 0, D - supp D. 2) , . , D = 0, vP (D) = 0. 3) , nP P = nP P P P ( nP . nP ) P . P F/k DivF. 4) 5) , - 0. D D , vP (D) D 6) 0. D deg D = : D = P, P F. ( D D, D D D–D 0), 0. vP ( D) deg P . P deg : DivF Div 0F deg D . DivF, 2.5.2. z. z (z)0 = , D . F, z : 0. Z ( N) – z; vP ( z ) P P Z (z) = - ( vP ( z )) P z; P N (z) = (z)0 – (z) = vP ( z ) P P z. ( ) - 17 , vP (z) = 0 (z)0, (z) (z) F \ {0} ( , x, y P, , , (z) . , div(z). - ) (xy) = (x) + (y), , PrincF 2.5.3. DivF. . k z F deg (z)0 = deg (z) = [F : k(z)], , deg (z) = 0. 2.5.4. 1) . D z Div 0F . PrincF – F/k – . D , F. D D . D – D = (z) , D– D D . [D]. 2) Pic0F = Div0F / Princ F . F / k. 2.5.5. . F = k(x) – k[x], v1,…, vn, 1 i n 1 v pi ( x ) ( z ) = vi vq j ( x ) ( z ) = j j m Ppi ( x ) – JF. , p1 (x),…, pn (x), q1 (x),…, qm (x) – > 0, 1,…, m – g(x) = p1 ( x)v1 ... pn ( x )vn , r = deg g(x), JacF h(x) = q1 ( x ) 1 ... qm ( x ) s = deg h(x), m , z = g(x) / h(x). : vi Pq j ( x ) – z, z, j v (z) = s – r P – z, s < r. u(x) – s–r z, s > r, , pi (x) n D+ = i 1 r–s qj (x), vu(x) (z) = 0. m vi Ppi ( x ) , D = j 1 j Pq j ( x ) D+ > 0, D > 0, deg D+ = deg g(x) = r, deg D = deg h(x) = s. (z)0 = D+ + (s – r)P , . , (z) = D . s < r, (z)0 = D+, (z) = D + (r – s)P . (z) = (s – r)P + D+ – D , deg (z) = 0. s > r, - 18 2.6. . 2.6.1. . D F/k L(D) = {z k F \ {0} (z) + D 0} {0} . 2.6.2. . L(D) – , D. 2.6.3. . 1) L(0) = k. 2) deg D < 0 3) D L(D) L(D ) deg D = deg D . L(D) L(D ) dim(L(D ) / L(D)) deg D – deg D. D = D+ D , D+ 0, D 0, dim D 1 + deg D+, L(D) . 4) D D 5) 1) F / k, L(D) = {0}. D 2) D 3) D, D – ( D, ). . D = D + (g) g L(D) , , L(D ), f L(D) P. vP (t) = vP (D ) = vP (D) + 1. , vP (xt) 0, . xt x Ker x , L(D). k 2.6.4. 2.6.5. 1) D – xt P vP (t), vP (x) vP (D). - dim k FP = deg P = deg D – deg D. dim L(D+) = dim(L(D+) / L(0)) + 1 D+ D dim D - F/k - deg D+ + 1. D D , vP (xt) > 0, L(D) L(D+). L(0) = k, . L(D). ; F, vQ (x) vQ (D ) = vQ (D), , Ker = L(D). : L(D ) / L(D) FP. deg 0. . t (xt)(P). , 4) dim L(D) FP, x , dim(L(D ) / L(D)) 5) D D+, dim(L(D+) / L(0)) deg D+ - k (xt)(P) = 0 1, , . vP (D ) = vP (x) , vP (t) = vP (D) Q, . 2.5.3. D =D+P L(D ) P. x vP (x) > 1 . : L(D ) . fg F . , 3). L(D ) *4) , (D). : 19 2) dim D 1; 3) dim D = 1. 1) 2) 2) 3) D D = (x) , dim D D D. x 1, F , . 1 , , x x L(D), x 0. L(D), deg D = deg D = 0. , dim D = dim D = dim 0 = 1. 3) 1) dim D = 1. x L(D), x , 0. (x) + D dim D 1. D = (x) + D. D = 0. - 0. deg((x) + D) = deg((x)) + deg D = 0, 1 (x) + D = 0, D = (x) = (x ) – 2.6.6. ( . ). g, F / k, , D dim D deg D 2.6.7. deg D + 1 – g , (2.7) (2.7) . . g F/k ( 2.6.8. L(D), z z ). . 2.5.5, 0, , v pi ( x ) ( z ) = vi vq j ( x ) ( z ) = (z) D, 0 = v pi ( x ) ( D) , 1 v (z) = deg g(x) i n; j = v (D), =0 1 j deg g(x) L( P ) = {g(x) , 0, D = P . – . 0 = vq j ( x ) ( D) , j - dim( P ) = + 1. m, . z = g(x) – . . , k[x] deg g(x) }, deg( P ) = , dim( P ) = deg( P ) + 1 – g, . + 1 = + 1 – g, , , , k(x) . g = 0. k= q, 20 3. 3.1. 3.1.1. . F/k D: F , x, y F, - F k 1) D(x + y) = D(x) + D(y), 2) D( x) = D(x). 3) D(xy) = xD(y) + yD(x), 2 3 D = D D, D = D D D . 3.1.2. ( 4) D( ) = 0 k, n n 1 5) D(x ) = nx D(x), 1 2 6) D(x ) = x D(x) x x ), F n– F, x 0, 0, . 3.1.3. ( ), DerF F/k F 3.1.4. . F / k, . k x– 1) DerF x x ( 2) 3) - : t , x (x) = 1. - x). F , x (t) DerF 0. F , . D , - x DerF D = D(x) x. , y– F / k, y 3.1.5. k[x], h(x) 0, ( ). = F = k(x) – , , g(x), h(x) – = (g(x) / h(x)) . , M– k (3.1) y (x) x. x (g(x) / h(x)) 3.1.6. F/k– F M (3.1) F. D: F D(xy) = xD(y) + yD(x). M, , x, y F . - 21 3.2. 3.2.1. . F/k– . F F 1 d:F F, : M F F : M, F , D= D: F d. M , - M F d D F F. F 3.2.2. M=F D: F D, F = D ( ). D: F F, F F– , , D= DerF F, (D, ) F D, D, =0 =0 D , DerF F F : , (DerF)*, F ( ( DerF, (DerF)*. ) F = 0, D = 0. . (D) = D, ( d. D, F. . D , ). , ), , . - , , F (DerF)*. . . 3.2.3. x F D DerF D, dx = 1 . F F S, . F (S) F dx . = D (dx) = D(x). S– (3.2) d x, x F, F (S) – (S) F , W– x F d(xy) x, y F k. F =F F F (S) xdy ydx, d(x + y) – dx – dy, / W. d( x) F dx, x F. dx (S) F F F (S) d. 22 . dx x F F - x. 3.2.4. : 3.1.1 , x, y F, k n 0 - 1) d(x + y) = dx + dy, 2) d( x) = dx. 3) d(xy) = xdy + ydx, , 4) d( ) = 0 n n 1 5) d(x ) = nx dx, 1 2 6) d(x ) = x dx 3.2.5. k, n– x F, x . 0, . k x F F. 1) x – 2) dx 0, 3) dx – , F - : F / k, F. F F = u dx, 3.2.6. 1) y : u F, dy = u dx D= x x (y) dy = = u F. x, dy = x, u dx = u 2) y– x, dx = u x (x) = u, dy dx dx t– = u dx = v dy dy . dx dy dy dx dy dx = dt = dt , dx dx dt dt ( ) dy dx dy = . dx dt dt 3.2.7. . - u, v , dy = x (y). (3.3) u = v x (y) = v 3) - dy = dx , F u dx = v x (y) dx, (3.3) (3.2), x (y) dx. dy = F, F. dy, 3.1.5 dx – d(g(x) / h(x)) = (g(x) / h(x)) dx. k(x) 23 3.3. – 3.3.1. k – . z . F, t– 0, P – F / k, t = z dt F/k F. vP ( ) = vP (z). 3.3.2. . t , 3.3.1 P. vP ( ) , vP ( ) = 0 . , . P, - , , ( )= vP ( ) P . P 3.3.3. . ( ) F, 0, , . F vP ( ) - . 3.3.4. ), ( x F F ) = (x) + ( ). , , , 3.3.5. . . D F (D) F/k ={ – F \ {0} ( ) D} {0}. k 3.3.6. . F. D i (D) = dim k F (D) D. i (D) 0 F (0), . D , . . 3.3.7. ( ) . F/k – . g. F (0) = g. W = ( )– k D - , 0, i (0) = dim k 3.3.8. . L(W – D) F / k. F (D), x x . , i (D) = dim k (W – D). 3.3.9. ( – ) D dim D = deg D + 1 – g + dim(W – D). , (3.4) (3.4) dim D = deg D + 1 – g + i (D). (3.5) 24 3.3.10. . 1) deg W = 2g – 2, 2) dim W = g, 3) deg D 2g – 1, – i (D) = 0 ( , , ): : dim D = deg D + 1 – g. , B– , , 3.3.11. deg B = 2g – 2 dim B ). . B– A, dim A 3.3.12. g, 1+ . , deg A 2g – 2, 1 deg A . 2 F = k(x) – (dx). Pp(x). p(x) – - k. k[x]. p(x) – d(p(x)) = p (x) dx, - dx = (1 / p (x)) d(p(x)) 1 1/x – 2 d(1 / x) = x dx, , vp(x) (dx) = vp(x) (1 / p (x)) = 0. , dx = x d(1 / x). 2 P , v (dx) = v ( x ) = 2. , (dx) = 2P . deg (dx) = 2 = 2g – 2 < 0, = z dx k(x). dim (dx) = 0 = g. z k(x). ( ) = (z) – 2P = (v (z) – 2)P + vP ( z ) P . P P – , , D= ( ) k[x], P – . P . vp(x) ( ) = vp(x) (z) k(x) (D). = z dx 0 = vp(x) (D). 2. 1, k(x) ( k(x) (D) - p(x) – z– . , = v (D), = {0}. P ) = {g(x) dx g(x) k(x) (D) k(x). , v ( ) = v (z) – 2 = deg z – 2 deg z z , k(x) (0) k[x], deg g(x) = {0}. 2, 2}. 3.4. P3.4.1. P– 1) FP = k P=t 2) 1 F / k, t – P. (2.5) P= 3) . z k P=k t P. P. : z = z0 = 0 + z1 t, z1 = 1 + z2 t, ………………… 25 zn = i k, zi P. + zn +1 t, n i i t (0 i n) , n + zn +1 t n +1 n (mod t n +1 z= + 1t +…+ nt z + 1t +…+ nt (3.6) , sn = + z – sn 1t P , +…+ nt n +1 ) (3.7) n . | z – sn |P (sn) z 1 c n 1 . P . , P z= ti i (3.8) i 0 P 3.4.2. u z– P , n . F, . u 0, n (3.6) t , z= i z = ut n - : ti . (3.9) i n ( (3.8)) t. , P0, n u z n = vP (z) P, - . 3.4.3. , . 1, t – (2.4), , , z P– F/k - , F z= i ti , (3.10) i n i FP ( , FP) 3.4.4. n 2.3.7, 0. n = vP (z) k , dz dt (3.10) , P, 0. ) i t i 1. (3.11) i n , n = z ( i i t(P) = 0, 1 dnz ( P) n ! dt n (3.12) 26 3.5. 3.5.1. . z P. F/k– P F , P– Res P, t (z) = TrP ( TrP : FP k– , t– t , 1 1) k, – (3.10). P– - , Res P, t (z) = 3.5.2. x, y F. F, 1. t, s – , P. = x dt = y ds - Res P, t (x) = Res P, s (y). 3.5.3. . 3.5.2 Res P ( ) = Res P, t (x) P. 3.5.4. 1) 2) 3) 4) 5) ( ). , F, k, z F Res P ( + ) = Res P ( ) + Res P ( ), Res P ( ) = Res P ( ), v P( ) 0 Res P ( ) = 0, Res P (dz) = 0, Res P (dz / z) = vP (z) deg P, z 0. 3.5.5. , ( ). F, 0, Res P ( ) = 0 P, , , Res P ( ) = 0. P 3.5.6. . 1) , z F F/k– vP (z) > 0. 1 1 z y P , , P– (1 z ... z n ) = - zn 1 n +1 = yt , 1 z , 1 zi . = 1 z 2) k(x) – = g(x) / h(x) – k[T]. , t– i 0 , P– k(x), vP (z) z 0, z = g1(t) / h1(t) n c0 + c1 t +…+ cn t (mod t c0, c1, ...,cn g1 (t) t, z = g1, h1 n +1 ) k, n h1 (t)(c0 + c1 t +…+ cn t ) (mod t n +1 ), 27 . 3) , 2 , z = (x + 1) / (x + x + 1) t = 1/x – 2 t +t 2 (x), 2 , x = 1/t P – 2 n (t + t + 1)(c0 + c1 t +…+ cn t ) (mod t : c0 0, c0 c1 c0 c1 c2 1, 1, ........................ ci ci 1 ci 2 , 3 i (t 3 j z=t+ j 1 vP (z) < 0 2 (x). 2 z = (t + t) / (t + t + 1). . t3 j 1) . n. n +1 ). 28 4. 4.1. * 4.1.1. . n = q – 1, – k 2 ={ , q 4.1.2. q, f q [T] deg f k k – 1}. k 1 1, T,…, T q . Lk 2 (f) = (f( ), f( : Lk f = 1}. {1,…, n} Lk = {f Lk n ,…, Ker , . (f) = 0, , Ker = {0} , , 4.1.3. . 4.1.4. . 1) d C = n + 1 – k, 2) . . C , q f )). , n – n ),…, f( , . q. deg f < k n, f = 0. - . [n, k]- C– C = (Lk) . – n k q. - : . C 1 G= ... k 1 1) x f , dC = n – k + 1. 2) 4.1.5. C, x q . ... 2 ... n ... ... ... 2( k 1) . n ( k 1) ... (f) x= 1 f w(x) = n – S . Lk, f S 0. deg f n – k + 1. (1), (T),…, (T , k 1 ) k – 1, w(x) n–k+1 n – k + 1, , dC = C. – - n = q – 1. CL (D, G) 4.2.1. F/ q – P1,…, Pn – D = P1 +…+ Pn. S– dC , q, 4.2. AG- 0, 1 : g; F/ q. 29 G– F/ 4.2.2. supp G x q, L(G) supp D = . , supp G i {1,…, n} x supp D = vPi ( D) = 0, vPi ( x) FPi = x(Pi ) Pi . q, deg Pi = 1. (x) = (x(P1),…, x(Pn)). n q : L(G) , , , q. 4.2.3. . AG- ), D – G, CL (D, G) = (L(G)). 4.2.4. . CL (D, G) [n, k, d]- : k = dim G – dim(G – D), d n – deg G. : L(G) CL (D, G), 4.2.2, 0 x L(G) x Ker (x) = 0. , , , (x) G vP (x) vP (G) P x(P1) = … = x(Pn) = 0. : P {P1,…, Pn} vP (x) vP (G) + vP (D) = vP (G – D), vP (D) = 0 , 1 i n vPi ( x) 1 = vPi (G ) + vPi ( D) = vPi (G D ) , vPi (G ) = 0, vPi ( D) = 1. , x (x) , Ker (G – D) , x L(G – D), , . Ker . - x L(G – D). = L(G – D). - , k = dim CL (D, G) = dim L(G) – dim L(G – D) = dim G – dim(G – D). , CL (D, G) d, . {0}, w( (x)) = d. n–d , L(G), x D x. Pi , P 1 vPi ( Pd x(Pi ) 1 i d vP (x) x(Pi ) = 0 vP (G) d+1 n. P. x L(G) Pi : {P1,…, Pn} vP (x) vP (G) = vP (G) + vP (Pd +1 +… + Pn) = vP (G – (Pd +1 +… + Pn)), vP (Pd +1 +… + Pn) = 0, i d vPi ( x) = 0 = vPi (G ) + vPi ( Pd 1 ... Pn ) = vPi (G ( Pd 1 ... Pn )) , vPi (G ) = 0 1 d+1 ... Pn ) = 0, i n vPi (G ) = 0, vPi ( Pd vPi ( x) 1 1= vPi (G ) + vPi ( Pd 1 x 0. ... Pn ) = vPi (G ( Pd 1 ... Pn )) , ... Pn ) = 1. (x) x i 2.6.3 3) (G – (Pd +1 + … + Pn)) , L(G – (Pd +1 +… + Pn)), , - 30 , , d deg(G – (Pd +1 +… + Pn) = deg G – n + d n – deg G. 4.2.5. . deg G < n, : L(G) CL (D, G) - : 1) CL (D, G) [n, k, d]- , , k = dim G deg G + 1 – g, d n – deg G, : k+d 2) 3) , , 2g – 2 < deg G < n, x1,…, xk – L(G), k = deg G + 1 – g. x1 ( P1 ) x2 ( P1 ) M= n + 1 – g. x1 ( P2 ) ... x1 ( Pn ) x2 ( P2 ) ... x2 ( Pn ) ... ... ... ... xk ( P1 ) xk ( P2 ) ... xk ( Pn ) CL (D, G). deg(G – D) = deg G – n < 0, – . – k+d n+1–g q = L(G – D) = {0} , 4.2.4 , - . 4.2.6. deg G < n F/ Ker – 4.2.5 k+d , , n + 1. g=0 ( , F= q (x) – ), k + d = n + 1, . AG- CL (D, G) . d * = n – deg G 4.2.7. dim G > 0 d * > 0. D, , 0 D 4.3. AG- , d = d* D, deg D = deg G dim(G – D ) > 0. 4.2.1. vPi (G D ) = 1, F (G – D) , i Res Pi ( ) ( ) = (Res P1 ( ),..., Res Pn ( )) . : , , C (D, G) 4.3.1. vPi ( ) CL (D, G). , F (G q. – D) n q , {1,…, n} Pi. 31 4.3.2. . AG- ), D C (D, G) = 4.3.3. . C (D, G) ( F (G – D)). d deg G – (2g – 2). deg G > 2g – 2 k = i (G – D) , 2g – 2 < deg G < n, 1) : C (D, G), (G – D) F vPi ( ) ( ) , ( )– Res Pi ( ) = 0 - {P1,..., Pn} vQ (G – D) = vQ (G) , 4.2.4 k = dim C (D, G) = dim d 4.3.1, Q 1. , Ker 2) Pi1 ,..., Pin n + g – 1 – deg G. k = n + g – 1 – deg G. vQ ( ) n : F (G – D) . 1 i – [n, k , d ]- k = i (G – D) – i (G) , ( G, G, F (G . Ker = – D) – dim , F (G). F (G) , = i (G – D) – i (G). C (D, G). d i {i1 ,..., in d } . n–d 4.2.4 , n d F (G ( D j 1 Pi j )) , 0. 3.3.10 3) n d deg(G ( D j 1 . d 3) Pi j )) = deg G – n + n – d = deg G – d deg G – (2g – 2). deg G > 2g – 2, , 2g – 2, – i (G) = 0 , - k = i (G – D) = dim(G – D) – deg(G – D) – 1 + g = = dim(G – D) + n + g – 1 – deg G 4) , , deg G < n, k = n + g – 1 – deg G. 4.3.4. n + g – 1 – deg G. deg(G – D) = deg G – n < 0 , , dim(G – D) = 0, d * = deg G – (2g – 2) . - C (D, G). 4.4. 4.4.1. , AG. vP ( ) P– 1, x – F/ F, , vP (x) Res P (x ) = x(P) Res P ( ). 0. q 1, – , 32 vP (x) 0, x x(P) = , vP (y) 1. vP ( ) = vP (z) 1, P = P q , , x= t– + y, P P, q. = z dt vP (y ) = vP (y z) = vP (y) + vP (z) , y z F. , Res P ( 4.4.2. . ) = Res P ( ) + Res P ( CL (D, G) ) = x(P) Res P ( ). C (D, G) , . . C (D, G) = CL (D, G) . 1) , – , P1, ..., Pn, vP ( , C (D, G) ) = vP (x) + vP ( ) , Res P ( ) = 0. F (G – D) x vPi (G ) = 0. vPi ( x ) n ( ) = i 1 2) P {1,..., n} Res Pi ( x ) = x( Pi ) Res Pi ( ) . (x); L(G). vP (G) + vP (G – D) = vP (G) + vP (G) = 0 vPi (G D ) = 1, vPi ( ) 4.4.1 i CL (D, G) . : n x( Pi ) Res Pi ( ) = i 1 Res Pi ( x ) = 4.2.4, 4.3.3 – Res P ( x ) = 0. P 3.3.9 dim CL (D, G) = n – (dim G – dim(G – D)) = = n – (deg G + 1 – g + dim(W – D) – deg(G – D) – 1 + g – dim(W – G + D)) = = n – (deg G + i (D) – deg G + n – i (G – D)) = i (G – D) – i (D) = dim C (D, G). , C (D, G) = CL (D, G) . 4.4.3. 2.4.1 vPi (t ) = 1 . t– y F, i: 1 i : , y(Pi ) = 1 1 i , . , vPi ( y ) = 0 Pi – , - y vPi ( ) = 1, Res Pi ( ) = 1 4.4.4. F, n, P1,..., Pn. 1 i n. = y d t / t. n. t 1 i n. , (4.1) (4.1). C (D, G) = CL (D, D – G + ( )). H = D – G + ( ). , supp H supp D 1 i n vPi ( H ) = vPi ( D) + vPi ( ) = 0. . , CL (D, H) 3.3.8 : L(( ) – (G – D)) F (G – D), x x . . 33 x L(H) = L(( ) – (G – D)) 1 i n vPi ( H ) = 0. vPi ( x) 4.4.1 Res Pi ( x ) = x( Pi ) Res Pi ( ) = x(Pi ). , (x ) (x) C (D, G), x 4.5. CL (D, H), x L(H), . , CL (D, H) = C (D, G). F (G – D) AG- 4.5.1. 4.2.1. AG- G , q (x) q + 1, x P CL (D, G), . , q+1 q (x) P x– q. 4.5.2. . C = CL (D, G) – n 1, k d. : 1) n q + 1. 2) k = 0 deg G < 0; k = n deg G > n – 2. 3) 0 deg G n – 2 AG- k = 1 + deg G - q d = n – deg G, , C . 4) C AG- . 1) 4.5.1. 2), 3) deg G < 0, dim G = 0. G – D < G, dim(G – D) = 0. 4.2.4 k = 0. 0 deg G n – 2, deg(G – D) = deg G – n , D CL (D, G) , - n , dim(G – D) = 0. dim G = deg G + 1 deg G L(G – D) L(G) , , 2<0 0 > 2 = 2g – 2, – k = dim G – dim(G – D) = deg G + 1. 4.2.6 d = n + 1 – k = n – deg G. , dim G = deg G + 1. , , deg(G – D) = deg G – n , deg G n–1 1 > 2g – 2, dim(G – D) = deg(G – D) + 1 = deg G – n + 1. , k = dim G – dim(G – D) = deg G + 1 – deg G + n – 1 = n. : deg G < 0 deg G n–2 k=0 1 deg G > n – 2 n – 1; 1 + deg G = k k=n n – 1; 1, . 4) 4.4.4. 4.5.3. . C = CL (D, G) – AG- q n, k, d. 0, 34 1) n q, 1,..., v1,..., vn q, , C = {(v1 f( 1),…, vn f( C n)) n 1} 1) deg f v2 2 v2 ... ... vn n vn 2 1 v1 2 2 v2 ... 2 n vn ... ... ... ... k 1 2 v2 v1,..., vn F= (4.2) . (4.3) k 1 n vn ... v1 1v1 v2 2 v2 2 1 v1 2 2 v2 ... ... ... ... ... ... k 1 2 v2 ... vn 1vn ... ... 1 1 n 2 v n 1 n 1 k 1 n 1 vn 1 0 , (4.4) 1 q \ {0}. 1 q (x). 1, k – 1} : k 1 1 v1 { 1,..., q [x], C M= q= , v1 1v1 k 1 1 v1 n = q + 1, - q : M= 2) f n D = P1 +...+ Pn. supp D. n Q q, P P , Q = P1. 1, - deg(Q – P) = 0 > 2 = 2g – 2, – dim(Q – P) = deg(Q – P) + 1 – g = 1 , , Q – P– 1 . (z) = P , z: P, P, P , 2 . (z) = P 3 . (z) = P , deg G n – 1, z = (x q q ) / (x z = 1 / (x z=x . q z– P=P . . Q – P = (z) z )=1+( , , deg G < 0, n q C= q. - ); ); q k=n ) / (x F, z F= k=0 q (z) . (z) = P. C = {0}. , - 4.5.2 , deg G = k – 1. deg((k – 1)P – G) = 0 > 2 = 2g – 2, , – , (k – 1)P – G = (u), t z u, L(G). t dim((k – 1)P – G) = 1 0 u F. , t t , , (k – 1)P – G – k – 1, k–1 (z u) = t (z) + (u) = t Q – t P + (k – 1)P G = t Q + (k – 1 – t)P G G. 35 t , u, zu, ..., z k 1 zu L(G). u 0 > 2 = 2g – 2, – . 0u q [z] + deg f 1 zu +…+ , k 1z k – 1. k 1 u = u( i= z(Pi ), vi = u(Pi ) 1 i , , , vi = z(P ) = z( ) = 0. , + 1z +…+ k 1z k 1 , , - ) = u f (z), q [z], deg f k – 1}. n. vPi (u ) Pi dim G = deg G + 1 = k L(G) : L(G) = {u f (z) f u , q. deg G = k – 1 f(z) z q G ) = 0, vPi ((k 1) P Pi = P , q . i 1 ( i z), z(Pi ) = n (u f (z))(Pi ) = u(Pi ) f (z)(Pi ) = vi f (z(Pi )) = vi f ( i ). (4.2). deg G < n, 0 j k – 1, (4.3). uz 2) P1 – Pn, Pn = P , 1), 4.2.5 j , L(G), j 1 , , vn j n) C, , n z q z. (v1 q. F, , - F= q (z) 1) (k – 1)P – G = (u ) 1 i F u n–1=q q= i { 1,..., = z(Pi ) , vi = u (Pi ) n 1}. v (u z (u z k 1 u , z u ,..., z q, )(P ) = k 1 k 1 u 1), q \ {0} q \ {0}. j 1 u= 1 L(G), vi = u(Pi ) = i: 0 1 i , - k 1 - n – 1. ) = v (u ) + (k – 1)v (z) = k – 1 – (k – 1) = 0, {u, z u,..., z – L(G). vi q k 1 u. u} \ {0} 1 i n–1 (u z )(P ) = 1. k–2 j v (u z ) = v (u ) + jv (z) = k – 1 – j > 0, j (u z )(P ) = 0. j k –1 1 i n–1 j j (u z )(Pi ) = u(Pi )(z(Pi )) = vi , j i . : j j k–2 j=k–1 j ((u z )(P1),…, (u z )(Pn)) = (vi ((u z k 1 )(P1),…, (u z k 1 )(Pn) = (v1 j 1, k 1 1 , , vn , vn j 1 n 1 , 0) ; k 1 1 n 1 ,1) . 36 (4.4). 4.5.4. F= x q (x) – , 1,..., – n q, Pi – i, n t (x) = (x i) . i 1 , y(Pi) = 1 F, t(x) – 1 i P1,..., Pn. n. y– 4.4.3 = y dt dx = y ( x )t ( x ) t t ( x) : vPi ( ) = 1, Re s Pi ( ) = 1 1 i n. 3.3.12 ( ) = (y) + (t (x)) – (t (x)) – 2P . 4.6. AG- 4.6.1. AG- C = C (D, G), supp D 4.6.2. 0, G1 – 1) G1 . t , supp G = , D = P1 +...+ Pn . d * = deg G – (2g – 2) – F / q. : supp G1 C , t– supp D = , deg G1 < deg G – (2g – 2) – t, (4.5) dim G1 > t, (d * – 1) / 2 , , 2) 0 t (d * – 1 – g) / 2, (4.5). t C t . G1, , , . , t - , z t (d * – 1 – g) / 2, F , vP1 ( z ) = g–t vPi ( z ) = 0 2 i n, G1 = (g + t)P1 + (z). 4.6.3. n q , f L(G) (4.5) . b = (b1,..., bn) 37 n [b; f] = b; (f) = bv f ( Pv ) , v 1 (f) = (f(P1), ..., f(Pn)), n q . L(G) q, (b, f) [b; f] C = CL (D, G) , n q C = {b 4.6.4. : c = (c1,..., cn) – , c e = (e1,..., en) – , e [b; f] = 0 f , w(e) t. n q ; n q , a v . n, , I = {v – (4.6) C . a = (a1,..., an) = c + e – ev, 1 L(G)} {1,..., n} ev 0} . {f 1, ..., f } L(G1); {g1, ..., gk} L(G – G1); {h 1, ..., hm} L(G); , 1 1 k (f g ) = (f ) + (g ) , , f g 4.6.5. –G1 – (G – G1) = –G L(G). . [a; f g ]x = 0, 1 k. (4.7) 1 x, 1 ( 1, ..., ). . (4.7) f= - f , (4.8) 1 f L(G1) f (Pv) = 0 , v I. . 4.6.6. f= . f 1 4.6.7. . N(f) = {v , 4.6.5 I N(f)). {1,..., n} f (Pv) = 0}. : 38 h ( P ) z = [a; h ], 1 m (4.9) N(f ) zv, v , , 1) f(Pv) = 0 supp G1 supp D = N(f). (4.9) (ev) v v N(f), N(f). 1 vPv ( f ) v, - , f Pv ) , L(G1 v N( f ) , Pv ) = deg G1 deg(G1 N(f) 0, v N(f ) . N(f) 2) deg G1. , h (ev) v N(f), ev – , (4.9). L(G), n [a; h ] = [c + e; h ] = [e; h ] = h ( Pv )ev . h ( Pv )ev = v 1 v N(f ) . 3) . (4.9). , n q b = (b1,..., bn) , [b; h ] = bv = 0 (bv) v N(f) – v N(f). h ( Pv )bv = [a; h ] = [e; h ] v N( f ) 1 m, [b – e; h ] = 0 b–e C . b–e w(b – e) bv – ev = 0 v 1 m. L(G), : deg G1 < deg G – (2g – 2) = d *, N(f) N(f). d * > w(b – e), dC 4.6.8. 1 . {h1,..., hm} – b – e = 0, . b = e. . ( 1, ..., ) (4.7). a. , (4.7) 2 . f= f . 1 3 . N(f) = {v f(Pv) = {1,..., n} f(Pv) = 0}. f (P ) . 1 4 . , (ev) v N(f) (4.9). (4.9) a. - 39 5 . e = (e1, ..., en), ev = 0 v N(f). , 6 . w(e) t. a. c = a – e. [c; h ] = 0 1 , m. a. , c C a - c. 4.6.9. (d * – 1 – g) / 2, , (d * – 1) / 2. - , (d * – 1 – g) / 2 . , (d * – 1) / 2. F , g =0 , , . 40 5. 5.1. k k(x). – - , . . 5.1.1. . F/k . F /k - F / k, F F k k. F ( F ), F /F ( . , 5.1.2. F / k, P F = k. F /k F / k , vP vP F, . e, , 1 e vP - F F / k. P= F, P , P , F/k F/k P. P P 5.1.3. P F ', . P , , , P P , P P. , P P , P F ', F ', ; P P, ; - e(P P); e(P P); , F / F, P, , P P: char k char k P P P P e F/ k F / F, , , P P. , . 5.1.2 e(P P). e(P P) = 1; e(P P) > 1; , e(P P) > 1 , e(P P) > 1 F / k. vP F = evP . P, P F /k P; F / F, P, F, P P P P P=P , P, ) ) k . , - k /k k k , - 41 F / F, , P P , - P; F / F, P P P e(P P) = [F : F]. F /F : , , , , P , 5.1.2 , FP = P P F / F. F F / F. F P 5.1.4. P F, F / F. P F' /P F, j: P. P P´ P P, - FP , j : FP : j FP FP j P P=P FP F=P F j ( FP ) . P =P FP P: P P, , j j ( FP ) , , FP , FP . FP P P 2.3.3 z P (z) = P (z) = z(P). 5.1.4. , FP / FP , FP . P P z(P ) = 5.1.5. : , - k /k . FP FP k k f(P P) = [ FP : FP ] P P. f(P P) deg P = deg P [k : k]. 5.1.6. F /k , . F /k F / k, P1, P2,…, Pr P. : r e( P i | P) f ( P i | P ) [ F : F ] . i 1 - 42 5.1.7. . P Con F F /k , F / F (P) = e( P | P ) P . P |P D DivF : Con F / F (D) = vP ( D ) Con F / F ( P ) , P Con F /F: DivF DivF . (x) . x F F F, x 0, (x) – x DivF [F : F ] deg D . [k : k ] deg(Con F / F ( D)) 5.1.8. D x DivF , DivF , F (x) = Con F / F ((x) F ) x. 5.1.9. , (T) = ( nT n + ). n 1T n 1 P 1) vP ( 2) vP ( n) = 0, vP ( i ) n) = 0, vP ( i ) +…+ F, 1T + – ( 0) > 0 (1 i n – 1) 0 (1 i n – 1), vP ( 0) < 0 F[T]. P P F , , (n, vP ( 0)) = 1; (n, vP ( 0)) = 1. F = F(y), , e(P P) = n = [F : F], P F. : i , (T) . F/k – y– (T), f(P P) = 1, F / F. 5.1.10. . v (f (x)) = d < 0. 1) n d 2) p = char k > 0 F = k(x) – , f (x) – n , p k[x] (T) = T – f (x) p (T) = T – T – f (x) d, d > 0. k(x). k(x). 5.2. 5.2.1. 1) P . F / k, P = P F /k F/k F . P , P |P 2) u1,…, un F F, , n P = P ui . i 1 F /F P. n, P – 43 3) , z1,…, zn . 5.2.2. F = FP – F F F/k – P , P– . F, , : P; a = a(P) – a (T) = F. P i ciT ci P, i ciT i (T ) = F [T ] . i , - F [T ] (T) = (T ) P [T], deg (T) = deg (T). P F/k F /k . 5.2.3. ( F / k. P ). ) F = F(y), ) (T) – ) (T ) = F /k y– F/k n, F . P F y P [T]. r i (T ) (T ) i i F [T ] . i 1 ) i – P [T], i (T ) = i (T) deg i (T) = deg i (T) 1 i P1,…, Pr F /k , r. P, , : 1) i ( y) Pi 2) f ( Pi | P ) , deg 1 i (T ) i r. , ) i : =1 ) 1, y,…, y 1 n 1 i r; – F /F P1,…, Pr, r; r 1) i (y) Pi, 2) e(Pi P) = 1 i i 1 i r P; P , , r Con F / F (P) = i Pi ; i 1 3) FPi = Pi F [T ] / ( i (T )) / Pi f(Pi P) = deg i (T) 5.2.4. . : n n 1 1) (T) = T + f n 1 (t)T + … + f 0 (t) – 2) F = k(t, y), y 1 i , , r. k(t); (y) = 0; 44 3) k, j, fj ( ) . j: 0 j [T], n – 1. , f j (t) - F / k, - . , y– f j (t ) = f j (t)(P ) = f j ( ) j n – 1, n 1 n (T ) = T + f n 1 ( )T + ... + f 0 ( ) k[T]; r 4) (T ) = (T ) i (T ) , i 1 k[T]. i , i (T ) i (T) = 1 , i r r. P1,…, Pr P 1) t – Pi, 2) i (y) Pi 3) e(Pi P ) = 1 P 1 i Pi 1 , r; , ) = e(Pi P ) v (t – ) = 1, vPi (t . t– 4) f (Pi P ) = deg i r; i (T) 1 Pi r; i deg i (T) = 1, f (P1 P ) = 1, 1 i r; , i {1,…, r}, , deg 1 (T) =1 - deg P1 [k : k] = f (P1 P ) deg P = 1 k F) : , k– 5) [k : k] = 1 , 6) deg Pi = f (Pi P ) = deg i (T) r = n, , . F; 1 i r. (T ) T– , ( ) = 0, P F / k, 7) t – P, 8) y – P, 9) deg P , = 1. k, P, F / k, , P. k, : t(P , ) = ; y(P , ) = ; 5.3. 5.3.1. F /k P F / k. F d(P P F , , P, P P), P Diff(F / F) = d ( P | P) P P P |P P, 45 F /k , F / F, - . . 5.3.2. ). P– F / k, P F /k , P. ) d(P P) e(P P) – 1, ) d(P P) = e(P P) – 1 , , char k = 0. 5.3.3. P , (T) – , . P char k e(P P). P , - , t– t F, d(P P) = vP ( (t)). 5.3.4. . F = F(y), y P, 1) d(Pi P) y– F P, vPi ( ( y )) 1 i r. 2) d(Pi P) = vPi ( ( y )) 1 i r 3) F /F 1 i vPi ( ( y )) = 0 P F , P (T) – P1,..., Pr – F /k , , 1, y,..., y n 1 - P. r, d(Pi P) = 0 , , e(Pi P) = 1, F / F. 5.3.5. 1) 5.3.1 P P P P : Diff(F F) , 2) P P 3) P P d(P P) = e(P P) – 1; d(P P) e(P P). , P , ( ). F / k ). F/k ( 2g , 2= , , - , 5.3.1, g ( : , x– , F/k: g g=1+ 5.3.8. k - [F : F ] (2 g 2) + deg(Diff(F / F)). [k : k ] , k(x) – F, , =k 1 deg(Diff ( F / k ( x ))) 2 ( ). F / k, k = - [F : k(x)]. g F = k(x, y) : g 5.3.9. F, F / F. 5.3.6. g) 5.3.7. P. . ([F : k(x)] – 1)([F : k(y)] – 1). F/k– (z dt) , t– : , z F. 46 (z dt) = (z) – 2(t) + Diff(F / k(t)), (t) 5.3.10. F / k. . F /F Aut(F / F), :F x F. , F , F / k, F /k , (x) = x Aut(F / F) = [F : F]. 5.3.11. . F /k F /k , F / k, P – P. i, j F / k, P1,..., Pr – {1,..., r} : 1) e(P) = e(Pi P) = e(Pj P); 2) f (P) = f (Pi P) = f (Pj P); 3) d(P) = d(Pi P) = d(Pj P); , , e(P) f (P) r = [F : F]. 5.3.12. Aut(F / F) . F /F , - . 5.4. 5.4.1. . F /k F = F k = F(k ) , 5.4.2. F /F 1) k 2) . F/k F k. F / k. F /k , F; k t– F, t k, - [F : k (t)] = [F : k(t)]; 3) t– F/k F = k(t, y) y F / k , F = k (t, y) k (t), 4) , k(t); F /F P , . e(P P) = 1 P, Diff(F / F) = 0. 5) 6) F /k g, D DivF deg(Con F 7) D DivF F / k. : / F (D)) : = deg D. y P F, F t– P F - , - 47 dim(Con F 8) W– / F (D)) = dim D. F / k, Con F / F (W) – - F /k . 9) FP FP P k - F. P=P k /k– F /F 10) FP k , P k k , F [F : F] = [k : k]. 5.4.3. . , k= q – , r– F/ 1) Fr / F – 2) q r 3) Fr / 4) 1, Fr = F qr q . qr : r; Fr; , qr P– F/ m q; F/ q. Con Fr / F ( P ) = P1 +…+ Pd, d= (m, r), P1,..., Pd deg Pi = m / d 1 i d. . 5.5. 5.5.1. ) k F = k(x) – , k= ) E = k(x, y), q , , n n q – 1). – n y = f(x) ) Q k(x), : , n char k k(x). , (n, vQ (f(x))) = 1 , f (x) – deg f (x), 5.5.2. k[x], n). . k– E. 5.5.1 E/k k(x). 5.5.3. : 1) E / k(x) – 2) 3) . (y) = y, n (T) = T – f (x) y k(x). P– k(x), 5.5.1 E/k n. n E / k(x) . k(x) - 48 e(P) = n rP d(P) = rP = , n – 1, rP (n, vP (f (x)) > 0, E / k(x) 5.5.4. f (x). : Diff(E / k(x)) = P P P |P deg P = P |P = [ E : k ( x)] n deg P = deg P = rP deg P; e( P ) n / rP P n 1 rP P n 1 rP deg P = rP = deg P = P |P (n rP ) deg P ; P 1 1 deg(Diff ( E / k ( x ))) = 1 – n + 2 2 g = 1 – [E : k(x)] + , , f (x) p(x), k(x) E / k(x). P ; P |P 1 1 e( P ) deg P = deg(ConE / k ( x ) ( P )) = e( P ) P | P e( P ) deg(Diff(E / k(x))) = 5.5.5. n 1 rP d (P)P = k[x] , r =1 , p(x). f (x). Pp(x) P (n rP ) deg P . P m, n, rp(x) = 1 - : g=1–n+ 1 2 (n 1) deg P = (n 1) P = (n 1) 5.6. 1 2 deg P 1 = P 1 (n 1)(m 1) (m 1) 1 = . 2 2 – 5.6.1. E = k(x, y) – ) . , y k– p > 0, k(x) – p y – y = f (x) ) Q – k(x). k(x), f (x), vQ (f (x)) k– , : E 0 (mod p). E/k– k(x). , , 49 5.6.2. . 5.6.1 – E/k - : 1) E / k(x) – p. (y) = y + v, v = 0, 1,…, p – 1. p (T) = T – T – f (x) y k(x). P k(x) 2) 3) E / k(x) - k(x) , . e(P) = p = [E : k(x)], f (P) = 1. f (x) = g(x) / h(x), g(x) h(x) – k[x], h(x) 0, h(x) = p1 ( x)m1 ... pr ( x )mr h(x) mi 1) Pi = Ppi ( x ) 1 i 0 (mod p) 1 i r. - k(x) r, d(Pi ) = (p – 1)(mi + 1). 2) deg g(x) – deg h(x) > 0 p, P . d(P ) = (p – 1)(deg g(x) – deg h(x) + 1). P– k(x), P deg P = deg P. Pi E / k, E / k, 5.6.3. Pi P, i {1,..., r, }. : Diff(E / k(x)) = d (P )P = d ( Pi ) Pi i = ( p 1)(deg g ( x ) deg h( x) 1) P = ( p 1)(mi 1) Pi i = ( p 1) (mi 1) Pi (deg g ( x) deg h( x) 1) P , i deg(Diff(E / k(x))) = ( p 1) (mi 1) deg Pi (deg g ( x) deg h( x) 1) = i (mi 1)deg pi ( x) deg g ( x) deg h( x) 1) = = ( p 1)( i = ( p 1)( deg pi ( x) deg g ( x) deg h( x) 1) = mi deg pi ( x ) i i = ( p 1)(deg g ( x) deg pi ( x ) 1) , i g=1–p+ p 1 deg g ( x) 2 5.6.4. deg h(x). deg pi ( x ) 1 = i p 1 deg g ( x) 2 1: m1 = ... = m r = 1, deg g(x) > deg h(x) deg pi ( x) , deg h(x) = : i g= p 1 deg g ( x ) deg h( x) 1 . 2 deg pi ( x ) 1 . i p deg g(x) – 50 5.6.5. 2: f (x) – k[x] m, k(x) p, P . - : deg(Diff(E / k(x))) = d(P ) = (p –1)(deg f (x) + 1), g=1–p+ p 1 ( p 1)(m 1) deg f ( x ) 1 = . 2 2 5.7. * 5.7.1. k– k, F = k(x, y) – q y + y = f (x) , : 1) deg f = m > 0 2) k– 5.7.2. p; q T + T=0 F. . 2) , q=p, k[x]. k. 5.7.1 p4.7.3. 1) s p > 0, k(x) – , F/k k(x). . p F / k(x) n ( /p ) . q (T) = T + T – f(x) y k(x). 3) F/k q. : - k(x) - k(x) P . F / k(x), e(P ) = q = [F : k(x)], , f (P ) = 1, , P , d(P ) = (q – 1)(m + 1). Q F / k, deg Q = 1. 5.7.4. F / k(x) Diff(F / k(x)) = (q – 1)(m + 1)Q . F/k g=1 1 1 (q 1)(m 1) (q 1)(m 1) q = deg(Diff ( F / k ( x))) [ F : k ( x)] = 1 . 2 2 2 5.7.5. x ( x )k ( x ) = P , y. (x) = ( x ) F = Con F / k(x) (P ) = e(P )Q = qQ . , vQ ( x ) = q. , vQ ( y ) vQ ( y q ) = qvQ ( y ) , 0. , vQ ( y q y) min{vQ ( y q ), vQ ( y )} 0. - 51 , , vQ ( y q vQ ( y ) < 0, , =Q k(x) y ) = vQ ( f ( x )) = e( P )vP ( f ( x)) = qm < 0. P . . Q y. Q Q F Q . P= vQ (y) < 0, q vQ (y ) = qvQ (y) < vQ (y) < 0. q q vQ (y + y) = min{vQ (y ), vQ (y)} = qvQ (y) < 0. , vQ (f(x)) = e(P)vP (f (x)) 0. , , y , , vQ ( y ) = m, 5.7.6. , (y) = mQ . 5.3.9 5.7.7. P P [T], . k(x), P , F / k, , d(P) = vQ ( F / k(x) 5.7.8. 1) r d(P) - vQ ( (y)) = vQ ( ) = 0 (y)) = 0. P. 1, y, ..., y - Q F z, , , z Q , z Q F Q Q P . k(x), P q 1 L(rQ ). Q P 5.3.4 P, 0– . z L(rQ ). k(x), P , P. P 2q)Q = y Q vQ (z) , x y ) = vQ ( f ( x )) = qm (dx) = 2(x) + Diff(F / k(x)) = = 2qQ + (q – 1)(m + 1)Q = ((q – 1)(m + 1) = ((q – 1)(m – 1) – 2)Q = (2g – 2)Q . , 0. Q . qvQ ( y ) = vQ ( y q , vQ (y) , Q = z P . P QP P - , q 1 z= zj yj , j 0 zj P k(x) P , zj P , k[x], . zj , P . - . q 1 z= aij xi y j , ai j k. (5.1) j 0 i 0 2) . , , i x y j j q–1 - 52 vQ ( x i y j ) = i q – j m = i1 q – j1 m = vQ ( x i1 y j1 ) . (i – i1)q = (j1 – j)m. , i = i1. , –(q – 1) j1 – j q–1 q , i, j, , iq + jm i , x y 0} r. vQ ( z ) i , x y j r L(rQ ) j 3) (5.2) i , (5.2) (5.1), iq – jm , j = j1 : vQ ( z ) = min{ i q – j m ai j , m, , ai j i, j. L(rQ ). vQ ( z ) 0, , , z 0. j x y, i 0, k. q–1 iq + jm i , j x y, L(rQ ) 5.7.9. j r, (5.3) (i, j) (5.3), - k. k. , k, q , + = f( ). q ( + ) + ( + ) = f( ) k, q , + = 0, q q T + T – f( ) = j), (T j 1 j – k. = 1,..., q Pj y ( Pj ) = q T + T – f( ) > 1. k P F / k, P, j= F, P, , deg Pj = 1. j k[T] k, f (P ) > 1 , deg P > 1. , - 53 6. k= – q q , F/ q n 5.1. – g. F/ An . F / q, , N = q + 1. , 0 - q , A1 N = N(F). r 1 . , Br , , , , F= q (x) r, h F/ An = 6.2. h (q n q 1 . F/ 1 g Z(t) n > 2g – 2, q n 0 . q. 1) . An t n = Z(t) = ZF (t) = 6.3. – t deg( D ) . D 0 |t| < q 1 ): Z(t) = L(t) = LF (t) – F / q. L (t ) , (1 t )(1 qt ) 2g 6.4. . 1) a0 = 1; g 2) a2g = q ; 3) a1 = N – (q + 1); g i 4) a2g i = q ai a1,..., ag. , L(t) = a0 + a1 t +…+ a2g t 0 i 6.5. g, 2g L- [t], : L 1,..., 2g – - , L , : 2g L(t) = i t) . (1 i 1 1,..., i 6.6. , 2g g+i =q 1 ( i - g. ). 1 i 2g 1/ 2 | i |=q . . 6.7. . 1 , |t| < q , 1 Z(t) = P F 1 t deg( P ) . 54 6.8. 1,..., . 2g Fr = F – F/ qr , L Nr = N(Fr) – Fr / 2g r LF (t) , , N=q+1– r Sr t r 1 (6.1) i i 1 Sr = Nr – (q + 1), L (t ) = L (t ) q. . qr i 1 1) F/ , 2g r i Nr = q + 1 – qr q : , r 1 2) a0 = 1 i ai = Si a 0 + Si 1 a1 +…+ S1 ai 6.9. 1 1 , , 6.10. g. N1,..., Ng, S1,..., Sg g i a 1,..., ag. . L(t) i a 2g i = q ai ( - i , g. ). F/ q , L- Fr / qr : N – (q + 1) 6.11. 2gq 1/2 2gq r/2 , N = q + 1 + 2gq 1/2 , , (6.2) . , q, ( . q : q q1/2 . 2 g 6.12. . , g, . r Nr – (q + 1) ). , q , - (6.2): N – (q + 1) g 2q 1/2 . 6.13. Br 6.1 K= , q Nr = r F/ q. d Bd . d |r r Nd . d r Br = d |r 2g Sr = r i , 1,..., 2g – , L i 1 r Nr = q + 1 Sr, : Br = r 1 1 r d|r r (q d d Sd ) . g>0 Br : qr r q q 1 2g q1/ 2 q1/ 2 q1/2 1 r 1 2 7g qr / 2 . r LF (t). 55 7. 7.1. 7.1.1. 1) 2) 7.1.2. ) . F/k A DivF . . char k 2, x, y x F, , y F = k(x, y), 2 f (x) – k[x], , ). x char k = 2, ( y . , 2 y + y = f (x) k[x], 2 deg f (x) = 3, y +y=x+ 7.1.3. = k(x, y) 1 , ax b a, b F– k, a k(x), char k y : , y = f (x) ) , g = 1; 0. 2, , F= F, 2 y = f (x) f (x) – k[x], , , : r f (x) = c pi ( x) , i 1 c k. Pi – k(x). P ) ) ) k(x), , : pi (x) k , n=2 v (f (x)) = 3 , F / k(x) – 1) F / k(x) – 2) i {1,..., r} char k 2; n = 2. , , : 2 k. vPi ( f ( x)) = 1, rPi = 1 , , e(Pi ) = 2 = [F : k(x)], Pi F / k, F / k(x), f (Pi ) = 1; Pi , deg Qi = f (Pi ) deg Pi = deg Pi . , d(Pi ) = 1. 3) v (f(x)) = 3, rP = 1 , , e(P ) = 2 = [F : k(x)], Qi – 1 i r, , 1 k; 56 P F / k(x), f (P ) = 1. F / k, Q P , deg Q = f (P ) deg P = 1; 4) P– , e(P) = 1, F / k(x). k(x), , P1,..., Pr , , d(P ) = 1. P1,..., Pr P , vP (f(x)) = 0, P rP = 2 , - k(x), Diff(F / k(x)) = Q1 +...+ Qr + Q . 5) F/k (n 1)(deg f ( x) 1) (2 1)(3 1) = = 1. 2 2 g= , F/k – 7.1.4. . F– k(x), p = char k = 2, y , F = k(x, y) F, 2 y + y = f (x) 2 y +y=x+ k[x] deg f (x) = 3 1 , ax b P a, b k, a k(x), P (7.1) 0. (7.2) ax + b k(x) ( (7.2)). , (7.1) : v (f (x)) = 3 < 0 3 0 (mod 2); (7.2) : v (x + 1 / (ax + b)) = 1 < 0 , F / k(x) , : 1) F / k(x) – 2) , . 1 0 (mod 2). – , - k. k(x) (7.1) P , - e(P ) = 2 = [F : k(x)]. f (P ) = 1. Q F / k, P , deg Q = f (P ) deg P = 1. , d(P ) = (p – 1)(deg f (x) + 1) = 4 3) P , Diff(F / k(x)) = 4Q . k(x) , . e(P ) = e(P ) = 2 = [F : k(x)]. f (P ) = f (P ) = 1. Q F / k, P, deg Q = f (P ) deg P = deg Q . (7.2) P 57 , d(P ) = (p – 1)(deg(ax + b) + 1) = 2, 2 d(P ) = (p – 1)(deg(ax + bx + 1) – deg(ax + b) + 1) = 2, Diff(F / k(x)) = 2Q + 2Q . 4) F/k g= p 1 2 (deg(ax 2 bx 1) deg(ax b) 1) = 1. , F/k – . 7.1.5. F/k , dim(0) = 1 = g 1 = y dx, = dx, k[x], deg f (x) = 3, 1 . char k = 2, f (x) = x ax b 1 z deg(0) = 0 = 2g – 2. char k 2, char k = 2, f (x) = (ax + b) dx, - k(x), = z dx, 5.3.9 (7.3) : (z dx) = (z) – 2(x) + Diff(F / k(x)), (z) = Con F / k(x) ((z) k(x) (x) = Con F / K ( x ) (( x ) ); k ( x) ) = Con F / k(x) (P ) = 2Q 2 (1): y = f (x), (f (x)) , . k(x) = P1 +...+ Pr – 3P , 2 2(y) = (y ) = (f (x)) = Con F / k(x) ((f (x)) k(x) ) = 2Q1 +…+ 2Qr – 6Q , , 1 (y ) = (y) = Q1 ... Qr + 3Q . , 1 1 (y dx) = (y ) – 2(x) + Diff(F / k(x)) = Q1 … Qr + 3Q 4Q + Q1 +…+ Qr + Q = 0. (2): (dx) = (1) – 2(x) + Diff(F / k(x)) = 4Q + 4Q = 0. (3): (ax + b) k(x) =P P , , (ax + b) = Con F / k(x) ((ax + b) k(x) ) = 2Q 2Q . , 1 ((ax + b) dx) = (ax + b) – 2(x) + Diff(F / k(x)) = 2Q + 2Q , , ( ) = 0. 4Q + 2Q + 2Q = 0. 58 7.2. 7.2.1. N F/ N 7.1.2 F= 2 (x, 2 + 1 + g[2 2 ] = 5. y), 2 y +y=x+ 2 y + y = f (x) Q– F/ 2. 1 , b x b 2 [x], 2, (7.4) deg(f (x)) = 3. , 2 (7.5) P 2 (x), deg(P) f (P) = deg(Q) = 1 , deg(P) = 1. 7.2.2. ) 3 Pb (7.4) 1. 2 (x) 1, , P , = b + 1. 2 (T) = T + T + x + , P0, P1, P . Qb Q , 1 x b , 2 (T ) = T + T + b. 2 : 1) b = 0, F/ (T ) = T(T + 1). 1, 2 2) b = 1 , Pb+1. (T ) = T + T + b Q b+1 ) F/ (7.5) P , 1. y 4 z = y + x. . 3 1) f (x) = x + x + 1. F/ Q , f (x) = x + bx + c, 2 (x, z) 2) f (x) = x . Q0 , , N = 3. Q0 F/ , 2, 2 P0. Q1 , N = 1. (T ) = T + T = T(T + 1) , 2 , 2 2 =0 - , P1. F/ ( 3 z + z = x + b1 x + c1). . Q0 2, 2 , 2 f (x) = x + x + bx + c, 3 (T ) = T + T + 1 – (T ) = T + T + 1 3 b, c 2 , N = 2. 2 =0 2 2 - - Pb+1. 3 , =1 1 2. 2, 2 F= Qb 1 , N = 4. 2 , Qb 2 1, Q1 , P0. F/ 2 =1 2, 2 2 (T ) = T + T + 1 P1. , - 59 3 3) f (x) = x + 1. 2 =0 (T ) = T + T + 1 – , Q0 F/ 2, 2 2 =1 (T ) = T + T = T(T + 1), 1, 3 4) f (x) = x + x. Q0 , Q0 4 . P0. Q1 2 P1. , 2 Q1 = F/ 2 , N = 3. =0 2 =1 Q1 , Q1 (T ) = T + T = T(T + 1) F/ 1, 2 , P0 , P1 - , N = 5. , F/ F= L 2 (x, 2 y), LF (t). N=5 2 3 y + y = x + x. 6.4 (7.6) : LF (t) = a0 + a1 t + a2 t 2 [t], a 0 = 1, a 1 = N – (q + 1) = 5 – (2 + 1) = 2, g a 2 = q = 2. 2 , LF (t) = 1 + 2t + 2t . = 1+i , Fr = F 1 i. 2 , LF (t) = (1 – t)(1 – t ). = 1 – i, 8- 7.2.3. 1 2 t= = = exp = 3 i 4 = 1 i 2 2. r 2r F/ Nr r r Nr = 2 + 1 – ( r r )=2 +1–2 2 r/ 2 Re( r r ) = 2 + 1 – 2 2r /2 cos 3 r. 4 : r 0 (mod 8) r = exp(0) = 1 r 1 (mod 8) r = exp 3 i 4 = r 2 (mod 8) r = exp 3 i 2 = i r 3 (mod 8) r = exp 9 i 2 = 4 2 r 4 (mod 8) r = exp(3 i) = 1 r 5 (mod 8) r = exp 15 i 2 = 4 2 r 6 (mod 8) r = exp 9 i =i 2 r 7 (mod 8) r = exp 21 i = 4 2 2 2 2 i 2 2 i 2 2 - 2 i 2 2 i 2 2 Re( r ) = 1, Re( r )= 2 Re( r ) = 0, Re( r )=2 Re( r ) = 1, Re( r )=2 Re( r ) = 0, Re( r )= 2 1/2 , 1/2 1/2 , , 1/2 , 2. 60 : 2r 1, , r 2 r Nr = 2 r 2 r 2 r 1 2 2 r/2 1 2 2 r /2 2, 6 (mod 8), , r 4 (mod 8), , r 0 (mod 8), ( r 1)/ 2 , r 1, 7 (mod 8), ( r 1)/2 , r 1 2 1 2 r 4 (mod 8) Nr = q + 1 + 2gq r 3, 5 (mod 8). 1 /2 . 0 (mod 8) Nr = q + 1 – 2gq 1/2 Nr = q + 1 + g[2q 1 /2 . r=1 ]. 61 8. * 8.1. * 8.1.1. 1) q– . , N: q +1 q, q2 . X q +1 –1 q2 . 2) Tr : q q, q2 . q q2 = q Ker Tr , 8.1.2. q2 H= Im Tr = - q q2 ( x, y ) , x– q2 H x q +1 +y q +1 = 1, q2 8.1.3. p = char q2 , n = q + 1, f(x) = x n y = f (x) , ) p n / Ker Tr Ker Tr = q. . H, y + 2 n q – 1, . q ) f (x) = (q + 1)x = x ( x) , q2 q2 (8.1) . q +1 + 1. (8.1) [ x] , n q2 q 0, . . f (x) , P– - f (x), (n, vP (f (x))) = 1. 5.5.1 , q2 8.2. H/ P 2 – H H/ q2 ( x) . * 8.2.1. q +1 q2 . q2 q2 ( x) , Q– deg P f(P) = deg Q = 1, 1, , P . P, deg P = 1. q2 . q2 1, - ( x) - 62 q +1 (T) = T +x q +1 – 1. : q +1 1) q2 = 1. P – , f (x) = x q +1 + 1, - , v (f (x)) = 1, . r = (u, v (f (x))) = 1, P H/ H/ q2 , P, H/ q +1 (T ) = T –1 q, . q ,..., q2 …, Q Q – q +1 q +1 + – 1. q2 , q +1 , q (T ) = (q + 1)T = T q – (T ) . H/ =1– q +1 , . q2 , . , , 1, q2 (T ) q+1 P. - 0, q+1 [T ] , q+1 , q2 , ( x)] , . q2 8.1.1 (T ) . , ( x) , f (P ) = 1. q2 1. q2 q +1 q2 deg Q = f (P ) deg P = 1. q+1 2) e (P ) = n = q + 1 = [ H : - Q , 1,... , 2 3 (q – (q + 1))(q + 1) = q – 2q – 1 H/ 3) = q2 . . , y P . z = y / x. z H= q2 1+ ( x, z ) (T) = T 1 x q +1 q2 1 x q 1 . 5.2.3 1 +1 x (T ) = T q 1 (T ) q +1 + 1. q+1 q2 , [T ] Q = 1+ z . q 1 8.1.1 q +1 , 1,..., Q , q+1 H/ q2 deg Q , , P , ,i =1 q+1 - ( x) q2 1 i f (P ) = 1 , q + 1. H/ q2 . , . - q+1 , 63 3 N=q +1 q2 H/ 8.3. , , 8.3.1. P1,..., Pq +1 – ( x) . , q2 , f (x) = x rPi = 1 deg Pi = 1 q2 1 . i H/ . q +1 +1 q + 1, , v (f (x)) = (q + 1), P ( x) . * H/ e(P ) = 1 q2 5.5.4 H/ q2 P1,..., Pq +1 rP = q + 1 , , ( x) . q2 q q(q 1) 1q 1 (( q 1) 1) deg Pi = q + (q 1) = . 2i 1 2 2 g = 1 – (q + 1) + 2 2 2 3 (q + 1) + 2gq = q + 1 + q (q – 1) = q + 1 = N. , . 8.3.2. 1,..., 2g – , L 2 LH (t), - 2g 2 N = (q + 1) + 2gq = (q + 1) – , i i 1 2g i = 2gq. | i |=q 1 i 2g. i 1 2g 2g 2gq = | i | i 1 Re i = 2gq i 1 2g , , (| i | Re i) = 0, Re i = | i | = q, Im i =0 , , i 1 i= q 1 i 2g. , 2g LH (t) = (1 + qt) . 8.3.3. , H/ H= q2 (u , v ) u q +1 +v q +1 q2 = 1. q2 , q , q + = 1, q +1 = 1, q c= . q + c = 0, q q q +c =( + c) = 0, q q q q )+ ( c + c= ( q q )= q +1 ( + q ) = 1. q2 , - 64 1 x= , , H= u v , u cv . u v y= ( x, y ) . q2 (u + v) q +1 q +1 x q +1 =1 q (u + v) (y + y) = q q = (u + v)( u + cv) + (u + v) ( u + cv) = q q +1 q q q q q q + c)u v + (c + )uv + ( c + = ( + )u + ( q +1 q +1 = u + v = 1. q c)v q +1 = , q y +y=x 8.3.4. deg f (x) = q + 1 q2 . (8.2) q p. Tr : H= q +1 q q, q2 T +T + . q2 . p ( x, y ) , q2 ( x) . - : 1) q2 H/ e(P ) = q = [ H : q2 ( x)] , Q – H/ q2 ( x) f (P ) = 1, H/ 2) q2 q2 P . ( x) d(P ) = (q – 1)(q + 2). , P , deg Q = 1. ( x) Diff ( H / q2 ( x)) = (q – 1)(q + 2)Q . 3) (x) = qQ , (y) = (q + 1)Q . 4) (dx) = (2g – 2)Q . i i 5) xy i 0, 0 j q – 1, iq + j(q + 1) r L(rQ ) . q2 6) q2 q q2 Q , H/ q +1 . , , q2 + = q +1 . 1, H/ N , q q , P q2 , x(Q P. H/ q2 , 3 N=1+q . , : , ) = , y(Q , )= . Q , 65 8.4. k * 8.4.1. d. n, H= q2 q , y +y=x 8.4.2. . P - , q 1 q , q2 ( x, y ) q +1 D= H/ q2 Q . r Cr = CL (D, rQ ). Cr. 3 1) 2) 3) Cr n=q . L(rQ ) L(sQ ) , L(rQ ) = {0} , r s, r < 0, 4) r > q + q – q – 2 = q + 2g – 2, 3 2 , Cr Cs. , Cr = {0}. 3 3 deg(rQ ) = r > 2g – 2 deg(rQ – D) = r – q > 2g – 2. , – : dim(rQ ) = deg(rQ ) + 1 – g = r + 1 – g; 3 dim(rQ – D) = deg(rQ – D) + 1 – g = r – q + 1 – g. Cr 3 3 k = dim Cr = dim(rQ ) – dim(rQ – D) = r + 1 – g – r + q – 1 + g = q = n. , Cr = 8.4.3. n q2 . , . Cr t= r 3 2 q + q – q – 2. : Cr Cq 3 (x ) = xq q2 q 2 r 2 . x q2 ( x) . q2 2 q2 ( , ) , q , v P , (t ) = e ( P , + = q +1 : | P )vP (t ) = vP (t ) = vP ( x , t ) = vP ( x P , ) = 1. D : H (t) = (t) = ConH / q2 ( x) (t ) q2 (x) = ConH / q2 (x ( x) q2 ) q2 ( x) = 66 = ConH / q2 P ) = ConH / (P ( x) q2 = , dt = d ( x q 2 q2 P P ( x) q2 = q2 P 2 3 q e(Q , x ) = dx, P )Q = D – q Q . , 2 (dt) = (dx) = (2g – 2)Q = (q – q – 2)Q . y = 1, 4.4.3 = ydt/t 4.4.4 Cr = C (D, rQ ) = CL (D, D – rQ + (dt) – (t)) = 3 2 = CL (D, (q + q – q – 2 – r)Q ) = Cq3 8.4.4. . : I = {n s q2 q 2 r z H, , (z) = nQ }. I(s) = {n I n s}. : 1) dim(n Q ) > dim(n – 1)Q z L(n Q ) \ L((n – 1)Q ) z H : (z) nQ z H : vP (z) P Q z H : vP (z) P Q z H : (z) = n Q n I. 2) dim(n Q ) – dim((n – 1)Q ) (n – 1)Q –n vQ ( z ) < (n – 1) vQ ( z ) = n deg(n Q ) – deg(n – 1)Q = n – n +1 = 1. dim(n Q ) , (z) , dim((n – 1)Q ) + 1 , n I dim(n Q ) = dim((n – 1)Q ) + 1. 3) {0} I(s) K L(0) L(Q ) , L(2Q ) 1 ... L(sQ ) dim(sQ ). , I(s) = dim(sQ ). 4) s 2g – 1 = q(q – 1) – 1, – I(s) = deg(sQ ) + 1 – g = s + 1 – q(q – 1) / 2. 5) 8.3.4, 5) , I(s) = {(i, j) 2 j q–1 i q + j(q + 1) s} . 67 8.4.5. . I (r ) , 0 1) dim Cr = 3 q3 3 r r q3 , I ( s) , q 3 2 2 q + q – q – 2. q3 r q2 q 2, 2 s = q + q – q – 2 – r. q –q–2 0. 8.4.3 Cr = Cs , - , 3 dim Cr = n – dim Cs = q – I(s) . 2) 8.4.4, 4). 3 n – deg(rQ ) = q – r. d 3 d=q –r 8.4.6. . Cr. T = {( , ) s = i q + j(q + 1), i 0, 0 j . + = q +1 }. q – 1, us = ( q3 q2 q 2 q2 r q – q – 2, Cq 3 3 2 q +q –q–2–r