Is there something to say about the irreducibility of polynomials and their derivatives?

Question

Is there a relation between the irreducibility of a polynomial and its derivative under certain conditions?

Answer 1

Very partial answer:

Any polynomial $g(X)$ over the rationals $\mathbb Q$ (or more generally a field of characteristic $0$) with degree $\ge 1$ has antiderivatives $f(X)$ that are reducible. In fact, given one antiderivative $f_0(X)$ you could take $f_0(X) - f_0(c)$ for any $c \in \mathbb Q$, which is divisible by $X-c$.

So having $f(X)$ reducible doesn't tell you anything about whether $f'(X)$ is irreducible.

Of course, if $f(X)$ is divisible by $q(X)^2$ for some polynomial $q$ of degree $> 1$, then $f'(X)$ is divisible by $q(X)$. If the degree of $f(X)$ is at least $3$, $f'(X)$ is then reducible.