I'm reading Proposition 14, page 15, Chapter I, taken in Lang - Algebraic number theory. It states that if $A$ is an integrally closed domain with field of fractions $K$, $L$ be a finite galois extension of $K$, $B$ be the integral closure of $A$ in $L$, $\mathfrak P$ be maximal ideal of $B$ lying over a maximal ideal $\mathfrak p$ of $A$, then $\bar B=B/\mathfrak P$ is a normal extension of $\bar A=A/\mathfrak p$.


The proof shows, actually, only that each finite separable subextension of $\bar B/\bar A$ is normal.
This implies that $\bar B$ is normal over $\bar A$? Why?



A proof in Lang - Algebraic number theory
Lang actually proves that for every $\overline{x}\in \overline B$ (no separability condition required), there exists a polynomial over $\overline A$ which has $\overline{x}$ as a root and splits into linear factors over $\overline B$. This yields that $\overline B$ is normal over $\overline A$.


