Abstract Algebra: Theory and Applications · Problem 17.5.25

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · polynomial-derivatives

Problem

Let

F

be a field and

f(x) = a_0 + a_1 x + \cdots + a_n x^n

be in

F[x]

. Define

f'(x) = a_1 + 2a_2 x + \cdots + na_n x^{n-1}

to be the derivative of

f(x)

. ((a)) Prove that

(f+g)'(x) = f'(x) + g'(x)

. Conclude that we can define a homomorphism of abelian groups

D: F[x] \to F[x]

D(f(x)) = f'(x)

. ((b)) Calculate the kernel of

D

\text{char}\, F = 0

. ((c)) Calculate the kernel of

D

\text{char}\, F = p

. ((d)) Prove that

(fg)'(x) = f'(x)g(x) + f(x)g'(x)

. ((e)) Suppose that we can factor a polynomial

f(x) \in F[x]

into linear factors, say

f(x) = a(x - a_1)(x-a_2)\cdots(x-a_n)

. Prove that

f(x)

has no repeated factors if and only if

f(x)

and

f'(x)

are relatively prime.