$LetFbe a field andf(x)=a_{0}+a_{1}x+⋯+a_{n}x_{n}be inF[x].Definef_{′}(x)=a_{1}+2a_{2}x+⋯+na_{n}x_{n−1}to be the derivative off(x). $

a)

$Prove thatConclude that we can define a homomorphism of abelian groupsD:F[x]→F[x]byD(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 polynomialf(x)∈F[x]into linear factors, sayf(x)=a(x−a_{1})(x−a_{2})⋯(x−a_{n})Prove thatf(x)has no repeated factors if and only iff(x)andf_{′}(x)are relativelyprime. $

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