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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 $[K(B):K(A)] \ge [k(B):k(A)]$?

This question should be 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.

Thanks!

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