Let F be a field and f(x)=a0+a1x+⋯+anxn be in F[x]. Define f′(x)=a1+2a2x+⋯+nanxn−1 to be the derivative of f(x).
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.
Suppose that we can factor a polynomial f(x)∈F[x] into linear factors, say f(x)=a(x−a1)(x−a2)⋯(x−an) Prove that f(x) has no repeated factors if and only if f(x) and f′(x) are relatively prime.
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