2022年抽象代数孟道骥版习题解答 .pdf

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1、Chapter33.1?1.Fp 0,F1) (a b)pn= apnbpn, ?n N ;2) (a1+ a2+ + ar)p= ap1+ ap2+ + apr;3) (a - b)p- 1=p- 1j =0ajbp- j - 1.1)(a b)pn=pnk=0Ckpnapn- k(b)k. Ckpnp?k =0k =pn.(pFp,px =0, ?x F.Ckpnapn- k(b)k= 0?k = 0k = pn.(- b)p=- bpp = 21 = - 1.(a b)pn= apnbpn, ?n N .2)1)(a1+ a2+ + ar)p= (a1+ a2+ + ar - 1)p+

2、apr= (a1+a2+ + ar - 2)p+ apr - 1+ apr= = ap1+ ap2+ + apr.3)(a - b)p= ap- bp= (a -b)p- 1j =0ajbp- j - 1(a - b)p- 1=p- 1j =0ajbp- j - 1.2.1) Z- 1/ 7 ;2) Z- 1/ 3 ;3) Z- 1/ 2 +- 1 ;4) Z- 1/ 1 +- 1 .1) Z- 1= a + b- 1|a, b Z . U,- 1=1,N ( ) = 1.= a + b- 1,a2+ b2= 1,= 1- 1.1, - 1U = 1,- 1,- 1, - 1 .7 Z- 1

3、- U.7 = (a+ b- 1).49 = (a2+ b2)N ( ).a2+ b21,7,49.a2+ b2=1,a + b- 1 U,48名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 1 页，共 16 页 - - - - - - - - - 49?a2+ b2=7,?a2+ b2=49,N ( )=1,?7.Z- 1UFD,7.Z- 1p.i.d.,7?Z- 1/ 7.?(a + b- 1) +7 Z- 1/ 7 .7(a + b- 1) +7 )=7(a + b- 1) +

4、7 = 0 +7 .7.2),3),4)7,3, 2 +- 1,1 +- 1Z- 1Z- 1/ 7 ,Z- 1/ 3 ,Z- 1/ 2 +- 1 ,Z- 1/ 1 +- 1.?Z- 1/ 77? Z- 1/ 33,Z- 1/ 2+- 15,Z- 1/ 1 +- 12.3.2.Z- 1/ 7 = Z7- 1Z- 1/ 3 = Z3- 1Z- 1/ 2 +- 1 = Z5- 1Z- 1/ 1 +- 1 = Z2- 1.4.Z= (Z ),(Z ) =Z31+ Z,Irr( Z, ).(Z ) = Z31+ Z(Z ) = (Z )Z31+ Z= (Z ).=(Z ) = (Z ).y =Z31+

5、Zy(1 + Z ) = Z3,Z3- yZ - y = 0,Z = (y)x3- yx - y = 0.Z.Irr( Z, ) = x3-yx - y.y?.fZ31+ Z=0.fm?(1 +Z )m,Zyyx3- yx - yx(y)x.5.KZpK?K ? Zp.K?= K -0.?K?,?n N ,an= 1.Zp.0Zp.6. x4+ 1 Qx.Q x4+ 1.4+ 1 = 0. ?1 = - 4.x4+ 1 = x4- 4= (x2- 2)(x2+ 2) =(x + )(x - )( x2- 42) = (x + )(x - )(x - 3)( x + 3).名师资料总结 - -

6、-精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 2 页，共 16 页 - - - - - - - - - 507. = Zp(x)Zp.1)zp- x Z ;2) zp-xzp- x( ).1)xZp(x)Eisenstein.2)p- x = 0, zp-x = zp- p= (z - )p.8.x2+ x +1 Z2x. x2+ x +1Z2( )4Z2( ).x2+ x + 1.(0)2+ 0 + 1 = 1,(1)2+ 1 + 1 = 1.x2+ x + 1Z2x. x2+ x+11- x2+ x

7、 +1=1 - .deg(, Z2) = 2, Z2( )22= 40,1, , 1 - .1 + = 1 -,1 + (1 - ) = - = , + (1 - ) = 1, (1 - ) = - 2= - 1 -= 1.9.C ,Irr( , Q):1) 1 +2;2)2 +3;3)1 +32;4)32 - 1.2) (2+3)2= 5+26, (2+3)3= 112+93, (2+3)4=49+ 206.(2 +3)4- 10(2 +3)2+ 1 = 0.x4- 10 x2+ 1 Qx.f (x) =ax3+ bx2+ cx+ d, a, b, c,d Q.a(112+93)+ b(5+

8、26)+ c(2+3) + d = 0. ?11a + c = 0,9a + c = 0, 2b = c, 5b + d = 0. ?a = b =c = d = 0,f (x)x4- 10 x2+ 1Irr( , Q).1)Irr(1 +2,Q) = x2-2x + 1.3)Irr(1 +32,Q) = x4- 10 x2+ 1.4)Irr(32 - 1, Q) = x12+ 3x8- 4x6+ 3x4+ 12x2+ 51),3),4)1).10.KF?K, F?F ( )F ( )F - ( ) = .“ ? ”, FIrr( , F ) = Irr( , F ) = p(x)?f : F

9、 ( ) F x/ p(x)F ( ) = x +p(x)? : F ( ) F x/ p(x)? ( ) = x +p(x):= - 1f : F ( ) F ( )名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 3 页，共 16 页 - - - - - - - - - 51 F ( )F ( ): ( ) = “ ? ”Irr( , F ) = p1(x),Irr( , F ) = p2(x)p1( ) = 0,p2( ) = 0.0 = (p1( ) = p1( ),p2(x)

10、 | p1(x)0 = - 1(p2( ) = p1( ), p1(x) | p2(x)p1(x) = p2(x), F.11.m N ,CQexp(2- 1/m ).= exp(2 - 1/m )xm- 1.xm- 1:, 2 , m= 1.k,(k, m) = 1.(k)m(k,m)= 1,m( k,m)= 1).Q ( )Q (k)Q , ( ) = (k).(k, m) = 1k.exp(2 - 1/m )?(m).12.KF K?,FIrr( , F ) =xn+a1xn- 1+ + an - 1x + an.- 1FIrr( - 1,F ) = xn+ an - 1a- 1nxn-

11、 1+ + a1a- 1nx + a- 1n.g(x) = xn+ a1xn- 1+ + an - 1x + anf (x) = xn+ an- 1a- 1nxn - 1+ + a1a- 1nx + a- 1nIrr( , F ) = g(x), n+ a1n- 1+ + an - 1+ an= 0a- 1n(- 1)n,(- 1)n+ an- 1a- 1n(- 1)n- 1+ + a1a- 1n- 1+ a- 1n= 0Irr( - 1,F ) | f (x)f (x),f (x) = f1(x)f2(x)g(x) = anxnf (1x) = anxnf1(1x)f2(1x)= anxn1

12、f1(1x)xn2f2(1x) = ang1(x)g2(x)(ni= Deg (fi(x),g(x).Irr( - 1,F ) = f (x).3.21.1) Q (2 +3):Q ;2) Q (2,3):Q;3) Q (2,35):Q;4) Q (32,36,324):Q;5) Q (2,6):Q (3);6) Q (2 +3):Q (3);名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 4 页，共 16 页 - - - - - - - - - 527) Q (2,3):Q (2

13、+3);8) Q (2,6 +10):Q (3 +5).2) Irr(2,Q)=x2-2.Q(2): Q =2. 1,2Irr(3,Q(2) = x2- 3.Q (2,3):Q (2) = 2. 1,3.Q(2,3) : Q = Q(2,3) : Q(2)Q(2 : Q =2 ? 2 = 4 1,2,3,6 .4) Irr(32, Q) =x3-2.Q(32) : Q =3.1,32,34.Irr(36,Q(32) = x3-6.(a1+ b1332 + c1334)(36)2+ (a2+b2332 + c2334)36 + a3+ b3332 + c3334 =0. ai, bi, ci Q

14、, i=1, 2,3. ?ai=bi=ci=0. i=1,2, 3.Irr(36,Q(32) 3.x3-636Irr(36,Q(32) =x3-6.Q(32,36) :Q(32) = 3.1,36,336 .Q(32,36)Q 1,32,34,36,312,324,336,372,3144 .324 Q(32,36).Irr(324,Q(32,36)=x-324.Q(32,36,324):Q(32,36)=1.Q(32,36,324)=Q(32,36).Q(32,36,324) : Q = Q(32,36,324) : Q(32,36)Q(32,36) :Q(32)Q(32) : Q = 1

15、? 3 ? 3 = 9.Q(32,36,324)Q 1,32,34,36,312,324,336,372,3144 .1)4, 3)6, 5)2, 6)2, 7)1, 8)2.2),4).2.1.2) 1,2,3,6 .4) 1,32,34,36,312,324,336,372,3144.1) 1,2,3,6 .3) 1,2,35,325,235,2325.5) 1,2.6) 1,2.7) 1 .8) 1,2.3.KFK : F = pK = F ( ), ?K - F.K - F ,F ? F ( ) ? KK : F = p,FK.K = F ( ).4.KF? , KF?deg(, F

16、)deg(, F ).Irr( , F )F ( )x?F (, ) : F = deg(, F )deg( , F ).F (, ):F =F (, ):F ( )F ( ):F =F (, ):F ( )F ( ):F .F ( ):名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 5 页，共 16 页 - - - - - - - - - 53F F ( ):F ?F ( ):F |F (, ):F ( )?deg(, F )|deg(, F ( ).Irr( , F ( )|Ir

17、r( , F ).deg(, F ( )deg(, F ),deg(, F ( )=deg(, F )F (, ):F =F ( ):F F ( ):F =deg(, F )deg( , F ).Irr( , F )Irr( , F ( )F ( )x.5.KF KFdeg(, F ):F (2) = F ( ).F (2) ? F ( ),F ( ) : F (2) = 1,F ( ) =F (2).? ? ? F ( ) : F (2)=F (2)( ):F (2) = deg(, F (2) .x2-2F (2).F ( ):F =F ( ):F (2)F (2): F .F ( ):F

18、 (2).F ( ) : F (2)1.F (2) = F ( ).6.a +b = 0,Q (a,b) = Q (a +b), ?a, b Q.a = b,.a = b,(a +b)2= a + b+ 2ab,ab Q (a +b).(a +b)3= aa + bb + 3ab(a +b),aa + bb Q (a +b)aa + bb = aa + bb-ba + ba=a(a -b) + b(a +b).a Q (a +b).b Q (a +b).a +b Q (a,b).Q (a,b) = Q (a +b).7.KFDF ? D ? K.D.D.?D -0 ,F.ni =1aii= 1

19、,:ni =1aii- 1= 1.,D.8.KCh K = p 0,Kpn(n N ).K: Zp = n, x1,x2, xnKZp.?K ,= 1x1+ 2x2 nxn.(iZp, i = 1,2, n)|K | = Pn.名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 6 页，共 16 页 - - - - - - - - - 543.31.QxQ-.1) x2+ 3;2) x5- 1;3) (x2-2)(x3- 2);4) x5- 3.3)3?(x2-2)( x3-2)2,32

20、0,321,322.E=Q(-2,2,320,321,322)=Q(2,32, ).(x2- 2)( x3-2)Q-= E : Q.E : Q.E : Q = Q(2,32, ) : Q(2,32)Q(2,32) : Q(2)Q(2) : Q.1. Irr(2,Q) = x2-2.Q(2) : Q = deg(2, Q) = 2.2.Irr(32,Q)=x3-2.Q(2,32)Q.2,32Q(2,32)Q?.deg(2,Q)deg(32,Q)?Irr(32,Q)Q(2)x?125? .Irr(32,Q(2) =Irr(32,Q) = x3-2.Q(2,32) : Q(2) =deg(32, Q

21、(2) = 3.3. Irr(Q)=x2+ x + 1.Q(2,32, )Q(2)?32,Q(2,32, )Q(2)?.deg(32,Q(2)=3.deg(Q(2)=2.deg(32, Q(2)deg(Q(2).Irr(Q(2)Q(2,32)x?.Irr(Q(2,32)=Irr(Q(2)=x2+x+1.?Q(2,32, ) : Q(2,32) = 2.1.2.3,E : Q = 2? 3? 2 = 12.1)Q(- 3),Q2.2)Q( )( 5,Q4.4)Q(53, )(5,Q5 ? 4 = 20.2.p? Zp( )Zp? ?.xp- Zp( )xKZp( )-.1)ZpZp Zp .xp

22、-Zp( )x.2) xp-p- =0. = p.xp- = xp- p= (x - )p?xp- p .K = Zp(, ).3) K : Zp( ) = Zp(, ) : Zp( ) = p.4) KZp( )- K: Zp( ) =p.p.KZp( ) |Zp()= idZp( ).K=Zp(, ), ( ). xp-Zp( )x? ( )xp- .? ( ) = . ?= idK. ?.KZp( )-1.名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 7 页，共 16 页 -

23、 - - - - - - - - 553.FKF.K = F ( ),Irr( , F ) = x2+ bx + c, (b,c F .x2+ bx + c = (x - )( x -c).cF ( ), KFx2+ bx + c.KF.4.Q (35)Q.Irr(35, Q ) = x3- 5 = (x -35)(x -35 )(x -352).(3) Q(35),Q (35)Q.5.K, EFF ? E ? K ,KFKE.KFKF? K, ?0 = f (x) F x,f ( ) = 0.F ? E ,f (x) E x.?EKE.p(x) E x,p(x)?p(x)K?p(x)K.?

24、, E?! :E ( )E ( )? ( )= , |E=idE. KF?g(x)=Irr( , F ),g( ) =0g ( ) =g( ( )=g( ) =0.g(x) =Irr( , F ).KFg(x) F xg(x)Kg(x)KK. ?KE.6.K, EF?F?E?K ,EFKF - (E ) = E. (E ) ?E .? E,p(x) =Irr( , F ). ? ( )p(x). ? ( ) E. (E ) ? E .E? (E ).- 1KF -.?- 1(E ) ?E ?E ? (E ).7.KFE1,E2E1,E2FKF - , (E1) = E2.“?” |E1E1 E

25、2F -.K. |E1. |E1(E1) =E2.K |E1. |E1E1E2. |F= idF.E1,E2.“?”Kf (x) F x?kf (x) E1x,f (x) E2x.E1, E2?1E1E2F -? 1|F=idF.1KK , |E1=1.名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 8 页，共 16 页 - - - - - - - - - 56 |F= 1|F= idF.KF - (E1) = |E1(E1) =1(E1) = E2.8.KFK : F + , KF

26、KF - ( ) = .“?”, KF, p(x) Fx.idF1: F ( ) F ( )1( ) =.F?F ( ) ?K,KFKF ( )? .K: F =K:F ( )F ( ):F + ,K: F ( ) + .KF ( )x.KF ( )x.3.3.31 : KK |F( ) =1.1|F=idF, KF -. ( ) = 1( ) = .“?”KFKF, F.g(x) = Irr( , F ), (g( )=g( ( ) =g( ) =0,g(x)Irr( , F )|g(x).g(x)F xIrr( , F ) = Irr( , F ), F.9.C0QC.C0QC0: Q =

27、 + .?C0,Irr( , Q ) = p(x).C0p(x)C0.C0Q.C0: Q = n,Eisensteinxn+1- 2Q x.21n +1C0,Q (21n +1) : Q = n + 1.C0: Q = + .3.41.FF xDD (f (x) + g(x) = D (f (x) + D (g(x);D (f (x)g(x) = D (f (x) g(x) + f (x)D (g(x);D (a) = 0, ?a F.DF x.:1) D (af (x) = aD (f (x) , ?a F, f (x) F x;2) D (f (x)m) = mf (x)m - 1D (f

28、 (x), ?f (x) F x,m N.1)?a F, f (x) F x,D (af (x) = D (a)f (x) + aD(f (x) = aD(f (x).2)m = 1.m = k,:名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 9 页，共 16 页 - - - - - - - - - 57D (f (x)k) = kf (x)k- 1D (f (x)m = k + 1:D (f (x)k +1) = D (f (x) f (x)k+ D (f (x)k) f (x)

29、= f (x)kD (f (x) + kf (x)k- 1D (f (x)f (x) = (k + 1)f (x)kD (f (x).2.DerF xF x.DerF x? (D1+D2)( f (x)=D1(f (x) + D2(f (x);F xDerF x? (f (x)D )( g(x)=f (x)D (g(x) .DerF xF x-F x-F x.(1):?f (x), g(x), h(x) F x, D DerF x,(f (x)D )(g(x) + h(x) = (f (x)( D (g(x) + h(x)= f (x)(D (g(x) + D (h(x) = (f (x)D

30、)( g(x) + (f (x)D )( h(x).(f (x)D )(g(x)h(x) = f (x)(D (g(x)h(x)=f (x)( D (g(x)h(x)+g(x)D (h(x)=(f (x)D )g(x) h(x)+(f (x)D )h(x) g(x).(2)Der F xF x-.?f (x), g(x), h(x) F x, D, D1, D2DerF x,(f (x)(D1+ D2) g(x) = f (x)( D1+ D2)g(x)= f (x)(D1g(x) + D2g(x) = f (x)( D1g(x) + f (x)( D2g(x)= (f (xD1+ f (x)

31、D2)g(x).(f (x) + g(x) D )h(x) = (f (x) + g(x)( Dh (x)= f (x)(Dh (x) + g(x)( Dh (x) = (f (x)D + g(x)D )h(x).(f (x)g(x)D )h(x) = (f (x)g(x)( Dh (x) = f (x)( g(x)D )h(x) .(1 D )f (x) = 1 (Df (x) = Df (x).(3)? : DerF x F x,?(D ) = D (x).?DerF x F x.?D0DerF x,:D0(anxn+ an- 1xn- 1+ + a1x + a0)= nanxn - 1+

32、 (n - 1)an- 1xn- 2+ + a1.,D0(x) = 1?f (x) F x, ?(f (x)D0) = f (x)?.? (D ) = 0,D (x) = 0,D (xn) = 0, D (f (x) = 0(?f (x) F x)D = 0,Ker?= 0.?.Der F xF x.3.Ff (x) F x.f (x)Kf (x) = (x -名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 10 页，共 16 页 - - - - - - - - - 58a1)(x

33、-a2) (x -an).:R(f, f ) =1jkn(aj-ak)2, (R(f, f )?f (x)( f ).)f(x) =ni=1f ( x)x- i.R(f, f ) =nj =1(ni =1f (j)j- i) =1j 0. xpn- aa = pn,xpn-a = (x - )pn.xpn-aF x?g(x), h(x) F x, 0 deg g(x),deg h(x) pn,xpn- a = g(x)h(x).g(x) =(x - )l1,h(x) =(x - )l2.(l1,p) =1,(l1,pn) =1.?u, v Z,ul1+ vl2= 1.g(x) F x,g(x)

34、 F .(- )l1F.l1F .pn= a F ,= ul1+ vpn= (l1)u(pn)vF .a =(pn -1)pFp.a Fp(l2, p) = 1.p|l1, p|l2,(pn, l1) = pl, 0 l n.?u, v Z,upn+ vl1=pl.pl=upn+ vl1=(pn)u(l1)v F, 0 l n.pn -1 F.a = (pn - 1)pFp.xpn- aF x.5.FK = F ( )f (x) = Irr( , F )KF -mm = red f (x).fKF -f ( ) = i.(iIrr( , F ).m = red f (x).6.Fp=0.F (

35、1,2, , n)F1,2, ,nF1) F (1,2, , n) : F (p1, p2, pn) = pn;2) F (1, 2, ,n)F (p1, p2, pn)-.1)n.n =11F?p1F.?f (x) F x,f (p1) =0.f (x) =0.f (p1)1.1F?f (x) = 0,p1F.xp-p1F (p1)x1.Irr( 1, F (p1) = xp-p1.F (1) :F (p1) = F (p1)( ) : F (p1) = p.n -1n.F (1, 2, , n) : F (p1,p2, pn)=F (1, 2, ,n - 1)(n):F (1,2, , n

36、- 1)( pn)F (1,2, , n- 1)( pn):F (p1,p2, ,pn- 1)( pn) = p ? pn- 1= pn.名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 11 页，共 16 页 - - - - - - - - - 592)? |F (p1, p2,pn)? |F=idF. (pi) =( (i)p? (i)p=pi. (i)p-pi=( (i) - i)p= 0. (i) = i.7.FqE = F (t)F:1) ?a F,EF -aa(t) = t

37、 + a;2) G = a|a F 3)K= x E |a(x) =x,?a F ,KEK=F (tq- t).1): |F= id,(f (t) = f (t + ),?f (t) F (t).?x F (t),(x) = 0 ?x = 0.?f (x) F (t), (f (t - ) = f (t).F -.2)?, , F,(t) = t + + , G.()( t) = t + + + = ()(t).0= = 0.- = 0.G.3)KE.(t + )q- (t + ) = tq+ q- t - = tq- t.F (tq- t) ? K .?f (t) F tf (t) K,f

38、(t + ) = f (t), ?F.g(t) = f (t) - f (0).?Fg(t) = 0.F?qt(tq- 1-1) | g(t),t(tq- 1-1) | f (t) -f (0).f ( t)- f (0)tq- tF t,f (t )- f (0)tq- tK.:f (t) F tq- t.?f ( t)g( t)F (t),f (t),g(t) F t.f (t)g( t) F (t) ?f ( t)g( t)=f ( t+ )g( t+ )?f (t + ) = f (t),g(t + ) = g(t) ?f (t), g(t) F tq- t ?f (t )g(t)F

39、(tq-t).K ? F (tq- t).K = F (tq- t).8.EFEF.“?”EF?EFF名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 12 页，共 16 页 - - - - - - - - - 60E.“?”1)Ch E = 0,Ch F = 0.F.2): E E (a ap).EE = Ep= (E ).(. |F: F F (b bp). |F. |FF (F ) =Fp. (E ) : (F ) = E : (F ) = E : F F : (F ).F= (

40、F ) =Fp,F: (F ) =1, (E ) : (F ) =E: F .E: F =n. 1, ,nEF. (1), , (n) (E ) (F ). 1, , nE, (1), , (n) (E ).a1, ,an (F )a1 (1) + + an (n)=0. ?b1, ,bn F,bpi=ai.ni =1bpi (i)=0.ni=1bpipi=0.bi, i ECh E =p,(ni=1bii)p=0.ni=1bii=0.1, ,nEFb1, ,bn0.a1, ,an0 (1), , (n). ?e (E ),?e E,ep=e.e =ep=(ni =1ei1)p=ni =1ep

41、ipi=ni =1 (ei) (i).e (1), , (n)?. (1), , (n) (E ).3.51.Q(2,3)Q.Irr(2,Q) = x2-2 = (x -2)(x +2). Irr(3,Q) =x2-3 = (x -3)( x +3).=2 + c3.c Q, c =2-23+3.c = 0,c = -2/3.Q(2,3) = Q( ), .2.Ex5- 2 QxE = Q( ).5E=Q(52, ). =+ c52, c Q.Irr(52, Q) = x5- 2.52,52522,523,524.Irr(Q) =x4+ x3+ x2+ x + 1.2,3,4.c =k-52-

42、52ik = 1,2, 3,4. i =1, 2,3,4.c1.=+52E =Q(52, ) =Q(52+).3.Z3( ) = F ,Irr( , Z3) = x2+ 1.F?= F - 0, F?F?F?F?= .1)Irr( , Z3) =x2+ 1Z3( ) : Z3 = 2.FZ3 1, .Fk1? 1 + k2? (ki=0, 1,2). F9.F?8.4.2= - 1, 3= - = 1,4= (- 1)2= 1. = 1, 2, 3 .F?.名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - -

43、 - - - - 第 13 页，共 16 页 - - - - - - - - - 612) FF?8.(2 )4=244= 1. (+ 1)2=2+ 2+ 1 =2.(+ 1)8= (2 )4= 1.(+ 1)i= 1, i = 1, ,7.= + 1.4.Zp(, )Zp, ZpF =Zp(p, p).1) Zp(, ) : F = p2;2) Zp(, )F.1) Zp(, ) : F =Zp(, ) : Zp(p,p)=Zp(, ):Zp(p, )Zp(p, ):Zp(p,p).xp-pZp(p, )x?pZp( )?.?f (x)Zp( )x,f (x) = 0,f (p) = 0.g

44、(x) Zpx,g(, ) =f (p)=0, Zp?.?pZp( )p.? xp-pZp(p, )x?.Irr( , Zp(p, )=xp-p.Zp(, ):Zp(p, )=Zp(p, )( ) : Zp(p, ) = p. xp-pZp(p,p)xpZp(p).xp-pZp(p, p)x?Irr( , Zp(p,p)=xp-p.Zp(p, ):Zp(p, p) = Zp(p,p)( ) : Zp(p,p) =deg(, Zp(p,p) =p.Zp(, ) : F = Zp(, ) : Zp(p, )Zp(p, ) : Zp(p,p) = p2.2) Zp(, ),Zp(, )=Zp(p,p

45、)( ).Zp(p,p),Zp(p, p, ) = Zp(p,p) = Zp(, ).Zp(, ) :F =p2.? Zp(p,p), Zp(, ),?cijZp,=i,jcijij(i,jcijij)- 1p= (i,jcijij)p=i,jcpij(p)i(p)jZp(p, p).p=(i,jcijij)- 1)p=(i,jcpij(p)i(p)j)- 1Zp(p, p).xp- pZp(p,p)x,.deg(, Zp(p,p) p.Zp(, ) : F = F ( ) : F = deg(, Zp(p,p) = p2.Zp(, )F.5.Fp =0,F (, )Fdeg(, F ) =n

46、, deg(, F ) = p.F (, ) : F .deg(, F ) = p.Irr( , F ) = (x - )p.F ( ),. F ( ).3.4.4:(x - )pF ( )x.F ( )( ) : F ( ) = p.F (, ) : F = F ( )( ) : F ( ) F ( ) : F = pn.名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 14 页，共 16 页 - - - - - - - - - 626.EFEFEF.“ ? ” F.F,?, E,F

47、 (, ) ? E,F (, )FF (+ x ) ? F (, )?x F .x1,x2F, x1= x2.F (+ x1 ) = F (+ x2 ).(, F (+ x1 ) = F (+ x2 ) = F (, ).eF.“ ? ” E = F( ),Irr( , F ) = p(x).F ? K ? E ,KFE.Irr( , K ) = p1(x) | p(x),K ( ) = E.KFP1(x),K? K .Irr( , K) | p1(x),E : K E : K .K = K.KEF.7.KFCh F = p = 0,KFKp= p|K FKK .8.KFCh F = p =

48、0,K0KFK0FK.:1) K0KKK02) ?K - K0, ?e N,peK0.1)?, K0, , .F (, )F-, - 1F (, ) ? K0K0K.? K -k0K0K0( )K0K0FK ( )FF.2)?K - K0,FEIrr( , F )Irr( , F )E:Irr( , F ) = (x - )pe(x - 1)pe (x - )pe.h(x) = (x - pe)( x - pe1) (x - pe)F x.KFpeK0.9.KFCh F = p = 0, K ,e 0peF ,F.KFKF1) KFred(Irr( , F ) = 1;2)KFF ;3)K0F

49、KKK0.名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 15 页，共 16 页 - - - - - - - - - 631)KF?e,Irr( , F ) = (x - )pe?red(Irr( , F ) = 1;2)KFIrr( , F ) = (x - ).F ;3)?K0,k,?K - K0, ?e N,peK0.(8KK0.10.Fp =0,E, KF?F?E?K,KFEFKE.“ ? ” KF.K, ?e 0,peF ? E,KE.E ? K, ?e 0,peF,EF.

50、“ ? ” ?K, ?e1 0,pe1E,?e20,(pe1)e2= p(e1+ e2)F.KF.11.KFCh F = p = 0,K0FKe 0E : F = K0: F pe.12.KFCh F = p = 0,KFE.13.KFKF - KF -?K, ( ) = i,iIrr( , F ).Irr( , F ),KF -= id.14.KFK0FKK0F.?K0, Irr( , F )K xIrr( , F ) = (x - )(x - 1) (x - m), i = j= i= j.i(1 i m)KF.K0F.名师资料总结 - - -精品资料欢迎下载 - - - - - - -