I have spent quite some time attempting to solve the following problem:
Let $A\to B$ be a finite homomorphism of Noetherian rings. We suppose that $A$ is a complete local ring, with residue field $k$. Let $k^\prime$ be the largest sub-$k$-algebra of $B\otimes_A k$ that is separable over $k$. Show that $A \to B$ decomposes into a finite étale homomorphism $A\to C$ followed by a finite homomorphism $C\to B$, with $C$ simple over $A$ (i.e., $C = A[c]$ for some $c\in C$) and $C\otimes_A k = k^\prime$.
So, as $B$ is an $A$-algebra, I want to start by writing $B = A[x_1,\cdots, x_n]/I$ for some ideal $I$, and then take $C = A[x_1, \cdots, x_n]$, hoping at the end that $n$ necessarily was $1$, but I have no idea why that might be the case. I really don't see another way to begin though, but if this is the wrong way to even start, I would appreciate any hints that you have to offer.