I have trouble understanding an argument in the proof of Proposition 2.5, p. 342, of Lang's Algebra. The setup of my question is the following:
Let $A$ be integrally closed in its quotient field $K$ and $B$ be its integral closure in a finite Galois extension $L$ of $K$, with group $G$. Let $p$ be a maximal ideal of $A$ and $\beta$ a maximal ideal of $B$ lying above $p$. Denote $\bar{B}=B/\beta$ and $\bar{A}=A/p$.
In his proof, Lang takes an element $\bar{x} \in \bar{B}$ that generates a separable extension of $\bar{A}$. He shows that $\bar{B}$ is normal over $\bar{A}$ and that $[\bar{A}(\bar{x}):\bar{A}] \le [K(x):K] \le [L:K]$, where $x$ is a representative of $\bar{x}$ in $B$. So far so good.
Then he says that "...this implies that the maximal separable subextension of $\bar{A}$ in $\bar{B}$ is of finite degree over $\bar{A}$ (using the primitive element theorem of elementary field theory)".
Question 1: How can we use the P.E.T. to see that? The statement of the primitive element theorem assumes that the underlying field extension is finite.
Question 2: Is $\bar{B}$ over $\bar{A}$ separable? Why is Lang referring to the "Galois group" of $\bar{B}/\bar{A}$?