Let $k \subsetneq K \subsetneq \bar{K}$ be a field extension, where $\bar{K}$ is an algebraic closure of $K$. (The fields are of characteristic zero). If $f \in k(x,y)[T]$ is an irreducible polynomial over $k(x,y)$, when is it irreducible over $\bar{K}(x,y)$? Actually, I am interested in the case where $k=\mathbb{C}(X)$, $K=\mathbb{C}((X))$, so $\bar{K}$ is the Puiseux field.
When I have asked that question I actually meant to ask the current question.