I have been trying to understand the structure theorem for etale maps for rings. Let $A\to B$ be a local homomorphism of Noetherian local rings which is etale. Then the structure theorem says that $B=A[T]_g/p(T)$, where $g\in A[T]$ is such that $p'(T)$ is a unit in $B$.
This theorem is given as Theorem 3.14 in Milne's book on etale cohomology. The way the proof proceeds, it seems that the degree of the polynomial $p$ is equal to field extension degree $[k(y):k(x)]$, where $y$ denotes the maximal ideal of $B$ and $x$ denotes the maximal ideal of $A$. This seems to be wrong since if we take $A$ and $B$ to be localizations of finitely generated algebras over an algebraically closed field, then this extension degree will always be 1, which would mean that $B$ is isomorphic to $A$.