Let $k$ be an algebraically closed field of characteristic zero, let $a,d$ be integers, and let $f\in k[x]$ be a separable polynomial of degree $d$.

Question: a) Is the affine plane curve $y^a=f(x)$ irreducible for all $a\geq 2$ and $d\geq 3$?

b) Same question as in a) but with $k$ of positive characteristic.