I am thinking about how to prove this fact: given a field extension $K\subseteq L$ such that $K$ is algebraically closed in $L$, and an irreducible polynomial $f\in K[X]$. Prove that $f$ is irreducible in $L[X]$.
I tried by writing down possible decomposition and relationship between coefficients of polynomials, but I got stuck.
Note that $K\subseteq L$ does not need to be purely transcendental, and $K$ is not algebraically closed per se. For instance, consider $K=\mathbb{R}$ and $L=\mathbb{R}((X))$.