高等代数多项式习题

1.

a b

x+

af(b) bf(a)

.

(A)f(x)g(x)=(f(x),g(x))[f(x),g(x)];(B)(C)(D)3.

f(x)=d(x)f1(x),g(x)=d(x)g1(x),

(f1(x),g1(x))=1;

d(x)=(f(x),g(x)).

u(x),∈K[x],

(f(x),g(x))=(f(x) g(x)u(x),g(x));

d(x)∈K[x],d(x)=f(x)u(x)+g(x)v(x),

f(x),g(x),p(x),q(x),d(x)∈K[x].

.

(A)(B)Q[x];(C)(D)5.

Cf(x)|g(x),q(x),r(x)∈C[x],

Qf(x)|g(x);

f(x)=g(x)q(x)+r(x),degr(x)<degg(x),

q(x),r(x)∈

u(x),v(x)∈C[x],f(x),g(x)C

f(x)u(x)+g(x)v(x)=1,

f(x),g(x)Q

Q

f(x)

g(x)

f(x),p(x)∈K[x],p(x)

Kk>1,

(f(x),f′(x))

6.

f(x)

C

(f(x),f′(x))

=f(x);

(D)f(x)=ap1(x)···ps(x),pi(x)Q

1

7.

.

(A)(B)

C[x]Q[x]

(f(x),g(x))=d(x),

d(x)∈Q[x];

f(x)|g(x)

C[x]f(x)|g(x);(C)f(x)Q[x]f(x)

C[x]

(D)f(x)Q[x]

f(x)C[x]

9.

n

2!

+...+

x.

(A)f(x)f(x)

(B)

q

1)

α 1,

f((A)f(x)Q

f(x)

g(x)C

(B)f(x)f(x)C

(C)f(x),g(x)

Q

f(x),g(x)

C

(D)C

x)

g(x),

Q

f(f(x)

g(x).

.

(1)f(x1,x2,···,xn)(2)i1(3)f(x≥i2≥···≥in;

1,x2,···,xn)

i1+i2+···+in;

(4)f(x1,x2,···,xn)

2

f(x)

g(x);

.

12.

1.2.

(x 1)2|ax4+bx2+1,

a=

.

g(x)=0,h(x)=0,g(x)h(x)

f(x)=g(x)q(x)+r(x),degr(x)<degg(x).

;f(x)h(x)g(x)

f(x)h(x)

.

3.

(f(x),f′(x))=x2+2,

f(x)=

,f(x)=

6.g(x)=f(x)

7.

f(x)

K

f(ax+b)

K

9.

f(x)=x4+4x2+5x+3,

f(x)=2x3 9x2+12x t

x3+px+q3+2i,

(

”

”

”

”)

2+i

3

;

C

.

11.