# Irreducibility of superelliptic curves

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.

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This is not suitable for MO. Not voting to close yet, as I don't know your background. In case you really need the answers, yes for (a) and for (b) if and only if the characteristic does not divide $a$. –  Felipe Voloch Apr 28 at 13:38
Dear Felipe Voloch, thanks a lot for answering my questions. I am new on MO and I am sorry that the questions are not suitable for MO. –  Robert Apr 28 at 13:45
Well, that's embarrassing. I am wrong on (b). It's always irreducible too. –  Felipe Voloch Apr 28 at 14:31
Dear Felipe, I was about to ask you about (b). If I am not mistaken, a) and b) have positive answers even only for $d>0$ (one can apply Eisenstein's criterion in $(k[x])[Y]$). –  Damian Rössler Apr 28 at 17:56