高等代数考研题选.pdf

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1、hlbfgwimtqyjonxkrgdsru(evpbf)_T[1.242*B(x−1)

2、Ax+Bx+1

3、Q2X6

4、ax+bx+2,

5、f(x),g(x)2X.
qf(x),g(x);86k>?=/
.7.kyW8

6、.
p(x)f(x)'s
/p(x)

7、f(x)jp(x)f(x)'s+k
/8.f(x)=x4+x3−3x2−4x−1,g(x)=x3+x2−x−1,'>/
(f(x),g(x))=.W[n,^;Æyvr<AgyrV2Z1.xC2X6{.
f(x);2X6{=

8、f(x),g(x),d(x)+6P{'.
xC;.

9、
u(x),v(x),&d(x)=f(x)u(x)+g(x)v(x),/
′4.

10、f(x)6P{'.
f(x)f(x)'$?k≥1.xCW8.′
p(x)f(x)'k
/


11、F606=f(x)∈F[x],p(x)F[x]GW8.
xC;8c0,&f(c0)=p(c0),<.(A)f(x)=g(x)(B)f(x)

12、p(x)1(C)f(x)=ap(x),a6=0(D)p(x)

13、f(x)2

14、.6{W8'.
f(x).(A)x2+3x+3(B)x3+3x+3(C)x2−3x−3(D)x3−3x−3YRZ[1.3sa,bb'&.
x+3ax+b2
/
2.<℄n:'a1,a2,···,an,s):'.
f(x)f(ai)=bi,=Ybi(<*'Di=1,2,···,n.3.3sa,bb'&.
x+3ax+b2
/
4.22s):z.
f(x)&f(x)+1l(x−1)?/f(x)−1l(x+1)?℄aV[1.33

15、F6f,g

16、∈F[x].xCg

17、f,

18、f.2.

19、F,F16qF⊆F1,f(x),g(x)∈F[x].(1)AgxC;F1[x]G2g(x)

20、f(x),<;F[x],(2g(x)

21、f(x).(2)Agf(x)5g(x);F[x]GG#qS#f(x)5g(x);F1[x]GG(3)Ag

22、f(x)6F{'W8.


23、x−1#qS#d

24、n.4.444-.
f(x1,x2,···,xn)=x1+x2+···+xnÆ(-.
'.
5..
f(x)

25、,g(x)G'Ag;[:.
a(x),b(x),&a(x)f(x)+b(x)g(x)=1.6.

26、Fw,6f(x)F{')7.
oa,n.A′n′gf(x)

27、f(x)#qS#;b∈F&f(x)=a(x−b),oGf(x)Æf(x)'$′′f(x)

28、f(x)Æf(x)?f(x).ab6=0.7.

29、f1(x)=af(x)+bg(x),g1(x)=cf(x)+dg(x)qcdAg(f(x),g(x))=(f1(x),g1(x)).8.x2x3xpAg.
f(x)=1

30、+x+2!+3!+···+p!;2X6{W8p29.(1)&AgA3?.
W8#'Eisensteinn1(2)n12iBT.'&C10.44s7.
f(x),&(x−1)

31、f(x)+1/(x+1)

32、f(x)−1.11.5Ag.
f(x)=x−5x+1;2X6Q{W812.3444

33、α,β,γ33x+3x−1=0'=sα+β+γ'D13.3s.
f=x+px−q'n
D(f)(0f'ÆD(f).n
'*-D(f)=(x1−x2)2(

34、x1−x3)2(x2−x3)2,x1,x2,x3f'8=p,q)14.YouareencourgedtoanswerthefollowingquestioninEnglish.Letα1,α2,···,αnbedifferentintegers.Provethatthepolynomialf(x)=(x−α1)(x−α2)···(x−αn)+1isirreducibleorasquareofsomepoly