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Let $g(y)\in \mathbb{C}[y]$, $ m\in \mathbb{Z}_{\ge 2}$. Are there some results on the irreducibility of $x^m-g(y)$ in $\mathbb{C}[x,y]$?

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    $\begingroup$ It is irreducible if and only if $g$ is not a $p$-th power for every prime divisor $p \mid m$. This is a special case of the Vahlen-Capelli theorem, which for an arbitrary field $F$ precisely determines the irreducible binomials $x^m-a \in F[x]$. $\endgroup$
    – Vesselin Dimitrov
    Commented Aug 23, 2014 at 23:23

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