I am attempting the part (b) and (c) of the following question from Judson:
I wrote this: $ker(D) = \left \{ f(x) \in F[x] : f'(x) = 0\right \} =\left \{f'(x) = a_{1} + 2a_{2}x + \cdots + na_{n}x^{n-1} : a_{1} = 0, 2a_{2} = 0, ..., na_{n} = 0 \right \}$
For part (b), I followed this answer. I am not sure why $char(F)=0$ implies $k$ of $ka_{n}$ is invertible for all $1\leq k \leq n$.
For part(c), I followed this answer. I understood it until they said "polynomial in $x^{p}$". What does that mean?