Introduce formal derivatives in F[x]. Let F be any field and let f(x) = a₀ + a₁x +· · ·a_ixⁱ · · ·a_nxⁿ The derivatives of f'(x) is the polynomial f(x) = a₀ + a₁ +· · ·(i .1)a_ix^i-1 +· · ·(n .1)a_nx^n-1 , where i . 1 has its usual meaning for i ∈ Z⁺ and 1 ∈ F. These are formal derivatives; no "limits" are involved here. 

a. Prove that the map D : F[x] → F[x] given by D(ƒ(x)) = f'(x) is a homomorphism of (F[x], +). 

b. Find the kernel of D in the case that F is of characteristic 0. 

c. Find the kernel of D in the case that is of characteristic p ≠ 0.