Abstract Algebra Polynomial Ring question I need some help

Abstract Algebra - Polynomial Ring question. I need some help on Part (b) & (c)

Let F be a ring and f(x) = a_0 + a_1x +...+a_nx^n be in F[x]. Define f\'(x) = a_1 + 2 a_2x +.... + na_n x^n - 1 to be the derivative of f(x) Prove that (f + g)\'(x) = f\'(x) + g\'(x) Conclude that we can define a homomorphism of abelian groups D : F[x] rightarrow F[x] by (D(f(x)) = f\'(x). Calculate the kernel of D if char F = 0. Calculate the kernel of D if char F = p. Prove that (f g)\'(x) = f\'(x)g(x) + f(x)g\'(x). Suppose that we can factor a polynomial f(x) F[x] into linear factors, say f(x) = a(x - a_1)(x - a_2)... (x - a_n). Prove that f(x) has no repeated factors if and only if f(x) and f\'(x) are relatively prime.

Solution

Here, it is given that (F(x), +,*) is a polynomial ring over the symbol x over the field F i.e., it is a set of all the polynomials of the given form f(x) of order n or smaller. Also, a homeomorphism D : F(x) --> F(x) of abelian groups is defined as D(f(x))=f\'(x). Clearly, D would not preserve the operation of multiplication, as D(f*g)=(f*g)\'=f\'g+g\'f which is not necessarily equal to f\'g\', it indicates that D is a homeomrphism of additive groups, as it preserves the operation of addition, given in (a) that (f+g)\'=f\'+g\' , i.e., D(f+g)= D(f)+D(g).

Now, the kernel of D as a homeomorphism of additive groups is defined as

ker(D)={f belonging to F(x) | D(f)=0, the zero element of F(x)}, as the zero of F(x) will be the constant polynomial

0_f = 0. It may be observed that as a constant function has its derivative to be 0, all constant polynomials in F(x) will be mapped to 0 = 0_f . It means that ker (D) will be the set of all constant polynomials i.e., functions of the form f(x) = k, where k is any constant in the field F. Also, it would be independent of the characteristic of F, as whatever characteristic F may be having, the set of elements of F mapped to 0 of F under D, will be the same.

Therefore, for both the parts (b) and (c), the ker(D) will be { f(x)=k belonging to F }.