Let $A\subset B$ be integrally closed local domains, $K(A),K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)]≥[k(B):k(A)]$?
This question is easy if $k(B)$ is generated by one element over $k(A)$, in particular, when $k(B)/k(A)$ is separable.
However, I don't know how to prove the general case. And I did not find this result in a book of commutative algebra.