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 $f \in k[T]$ is an irreducible polynomial over $k$, when is it irreducible over $\bar{K}$?
(An easy counter-example: $\mathbb{Q} \subsetneq \mathbb{R} \subsetneq \mathbb{C}$, $f(T)=T^2+1$).
Actually, I am interested in the case where $k=\mathbb{C}(X)$, $K=\mathbb{C}((X))$, so $\bar{K}$ is the Puiseux field.
Over an algebraically closed field, the only irreducible polynomials are linear polynomials (since every nonconstant polynomial has a root, and hence a linear factor). So the only irreducible polynomials over $k$ that are still irreducible over $\bar{K}$ are linear polynomials.