03 - XXX P64 11, 12, 16, 19, 21. 1 1203 P45 − P54 , øƒ...

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Unformatted text preview: XXX P64 11, 12, 16, 19, 21. 1 1203 P45 − P54 , øƒ ª ¸ : ª » ‚ ƒ x* 3:30-5:30 1 1108 ª x » ‚ 1 xƒ‚»ª 4i ii ) f ( X ) = an X n + + a1 X + a0 , c ∈ F , f (X ) Fƒ‚»ª* f (c) = an c n + + a1c + a0 , f ( c ) = 0, c f ( X ) c f ( X ) = 0ƒ‚»ª c 2 i) ii ) c f (X ) m f ( X ) = ( X − c ) q( X ) + f ( c ). f ( c ) = 0 ⇔ ( X − c ) f ( X ). ⇔ f ( X ) = ( X − c ) q( X ) f ( X ) = ( X − c ) g ( X ), ( g ( X ) ∈ F ( X ), c ∈ F , g ( c ) ≠ 0, m ≥ 1) 2 c f (X ) m m =1 c 1 31 F1 1 n1 ª ‚» (h‚* n1 f ( X )1 F1 ). proof : a1 f ( X ) , h‚»ª* ∃ f1 ( X ) f ( X ) = ( X − a1 ) f1 ( X ) a2 a1 ≠ a2 f ( X ) = ( X − a1 )( X − a2 ) f 2 ( X ) f ( X ) = ( X − a1 ) ( X − am ) f m ( X ) = ( X − a1 ) ( X − am ) » 1‚ h‚* ,∴ ∑ni ≤ n. i =1 m n1 nm f ∗ m(X ) 3 X X FX f ( X ) ∈ F [ X ], deg f ( X ) = n, X > ng • ﻪ , X f ( X ) = 0. f (X ) 4 f ( X ), g ( X ) ∈ F [ X ], deg f , deg g < n. n• ﻪ c1 , , cn , f (ci ) = g (ci ) f ( X ) = g ( X ). X h( X ) = f ( X ) − g ( X ). X c1 , , cnX h( X •)g ﻪ* ,∴h( X ) = 0. 4 §1- 3 d Ý 1 1) f , g ∈F [ X ] h f, hg g ( X )ø L ƒ ¸ ª h( X ) f (X ) h ( X ) ∈F [ X ] 2)1 d ( X ) ∈F [ X ] 1 f 1 g1 1 f 1 gà 1 d ( X )1 f ( X )1 g ( X )øƒ L * ¸ ª , . f X gà b ( f , g ). 5 ( f , g ) = ( r, g ). f ,, X( X proof : X ( f , g ) f , ( f , g,)ggr ∈ F [ f ,]g ) r , 1 X ( f , g ) ( r , g ); X ( r , g ) ( f , g ). f = gq + r, f =gq1 + 1 , r deg r < deg g 1 g = 1q 2 + 2 , r r deg r2 < deg r 1 r = 2 q3 + 3 , r r 1 rs −2 = rs −1qs + rs deg rs < deg rs −1 rs −1 = rs ⋅ qs +1 ∴( f , g ) = ( g , r ) = = ( rs −1, rs ) = crs 1 6 1 2.5 g(X) x3+ 2x2 -3 x3 + x2 - 2 x x2 +2x -3 x 2 + x -2 r2(x)= x -1 q2(X) =x +1 f(X) x4+ x3- x2- 2x+ 1 x4+2 x3 - 3x - x 3 - x 2 +x + 1 - x3 - 2x2 + 3 r1(x)= x2 +x -2 =(x-1)(x+2) q1(X) = x -1 1 f = gq1 +r , 1 ( f, g ) = r2(x) = x -1 g =r q2 +r2 . 1 7 r2 = g −r q2 = g −( f −gq1 ) q2 . 1 = − 2 f +(1 +q1q2 ) g .=( f, g ) q f =gq1 + 1 , r g = 1q2 + 2 , r r rs −2 = rs −1qs + rs , deg rs < deg rs −1 rs = rs −2 − rs −1qs = ( rs −4 − rs −3qs −2 ) − ( rs −3 − rs −2 qs −1 ) qs = = f 1 g xë•»ª* 1 1 21 f , g ∈ F [ X ], 1 ( f , g ) = d ( X )1 ,1 u, v ∈ F [ X ], 1 uf + vg = d Bezout1 . 8 §1- 4 f , g ∈F [ X ], f ( X ) g( X ) X ( f , g ) =1 3X( f , g ) = 1 ⇔X u, v ∈ F [ X ] X uf + vg = 1. proof :⇒X Bezout X X d ( X ) = 1X ⇐X ( f , g ) = d ( X ), X d ( X ) f , d ( X ) g , ∴d ( X ) 1 ∴d ( X ) = 1. X X 9 X 4X i )X f gh, X ( f , g ) =1, X f h. ii )X f1 g , f 2 g , X ( f1, f 2 ) =1, X f1 f 2 g . iii ) X ( f , g ) =1, ( f , h ) =1, X ( f , gh ) =1. proof : i ) u f + v g =1, u f h + v g h = h → f h. h ii ) g = f1h1, f 2 f1h1, ( f1, f 2 ) =1, ∴ f 2 h1 ∴h1 = f 2 h2 ,∴g = f1 f 2h2 , f1 f 2 g . iii ) X uf + vg = 1, sf + th ª = 1ú, h »€ uf ( sf + th ) + sfvg + vtgh = 1 ∴( f , gh ) = 1. 10 §2- 1 •è » * î ª 3 +n ª f» • 3 +n f 2 1 F F f = gh, ( g , h ∈F [ X ] f ) 1 2 x − 21 ª Qè * »• , 1 C1 R1 . ; x +11 R1 11 X 1 X p ( X ) ∈ [ X ]X F (i )X pX fX , Xp f . (ii )X p f1 f 2 , X p f1 X p f 2 . proof : (i ) p( 󀻪 p( 󀻪 ( p, f ) p ( p, f ) =1 p, f p . ∴ p, f ) = p → p f ( (ii )1 p1 f1(ó* ª»€ 1 p1 f11 p f 21 (i ) p f 11 12 1 X Xß • ø‚»ª* 61 F [ X• ]ø‚»ª* . f ∈ F [ X ]• ø‚»ª* f = p1 p2 ps . pi X proof : (i )Xß f •‚»ª* f , f = p1 f = f1 f 2 deg f i < deg f ª‚ •»* f1 f 2 ª ‚ •» ∴ f = f1 f 2 = p1 ps . (i = 1,2) 13 (ii )8 ç•»ª* ps q1 qt . f = p1 ps = q1 qt i = t, 1 ps qi qt , qt ps qt ps ∴8ç•»ª , ps = qt . ∴ p1 ps −1 = q1 qt −1 8ç•»ª p1 = q1, , ps = qt . f n1 ns = cp1 ps piH 14 1f ni = ∏ pi i =1 s s g mi =∏ pi i =1 s ni , mi ≥ 0 1 ( f , g) min( ni , mi ) = c ∏ pi i =1 »•æ ˆ* [ f , g ] = c′∏p ˆª* i= 1 s max( ni , mi ) i c′∈F . 15 exp1 1 f ( x) = x + 2 x − 3x − x + 1, g ( x) = x + 2 x − x − 2 , 5 4 2 3 2 ( f ( x), g ( x)) = u ( x) = v( x) = u ( x) f ( x) + v( x) g ( x) = ( f ( x), g ( x)). 86 1 5 16 ¨ß ¨ß 1 ô ¨€»ª* ô ¨€»ª* 11 a ∗ a 1 e, a ∗ e1 a. −1 a ∗a = e ∗ (a ∗ a ) = (a ) ∗ a ∗ (a ∗ a ) −1 −1 −1 −1 −1 −1 −1 −1 = (a ) ∗ e ∗ a −1 = (a ) ∗ a −1 −1 −1 −1 = e. . a ∗ e = a ∗ ( a ∗ a ) = ( a ∗ a ) ∗ a = e ∗ a = a. 30.1 F1 , F2¨€»ª* ô F1 ∪ F2¨€»ª* ô 17 exp2 X f ( x) = x + x − , 1 4 4 1 4 2 1 2 g ( x) = x + x − x − , 1 4 3 1 4 2 1 2 1 2 à ( f ( x), g ( x)) = u ( x) = u ( x) . v( x) = u ( x) f ( x) + v( x) g ( x) = ( f ( x), g ( x)). v( x) = . 18 g ( x) q2 = 1 4 13 4 13 4 f ( x) 1 2 1 2 x+ 1 4 x + x − x− x −1x 4 1 4 2 1 4 1 4 2 14 4 14 4 x − 1x− 1 4 2 2 1 x −4 1 4 1 4 r2 = − x − x + x − 1 q1 = 2 13 12 1 x + 4x − 2x − 2x x−1 13 32 1 1 − 4x + 4x + 2x− 2 13 12 1 1 − 4x − 4x + 2x+ 2 1 4 2 = − 1 ( x + 1) 4 r1 = x − 1 2 = ( x − 1)( x + 1) ∴ ( f , g ) = ( x + 1) = −4r2 ( x) 19 f = gq1 +r , 1 g =r q2 +r2 . 1 r2 = g −r q2 = g −( f −gq1 ) q2 . 1 = − 2 f +(1 +q1q2 ) g . q ∴ ( f , g ) = ( x + 1) = −4r2 ( x) = 4q2 f − 4(1 + q1q2 ) g v( x) = −4(1 + q1q2 ) = −(4 + x − 1) = − x − 3 2 2 ∴ u ( x) = 4q2 = x + 1; (u , v) = 1 20 ...
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This note was uploaded on 03/25/2010 for the course MATH 40 taught by Professor F.yu during the Spring '05 term at Tsinghua University.

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