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tobethederivativeof\begin{array}{l}{\text { Let } F \text { be a field and } f(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n} \text { be in } F[x] .

 Let F be a field and f(x)=a0+a1x++anxn be in F[x]. Define f(x)=a1+2a2x++nanxn1 to be the derivative of f(x).

a)

 Prove that  Conclude that we can define a homomorphism of abelian groups D:F[x]F[x] by D(f(x))=f(x)


b)Calculate the kernel of D if char F = 0.

c)Calculate the kernel of D if 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)F[x] into linear factors, say f(x)=a(xa1)(xa2)(xan) Prove that f(x) has no repeated factors if and only if f(x) and f(x) are relatively  prime. 

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