# Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$.

For a prime $b$ of $B$, let $e_b$ be its ramification index and $f_b$ its residue degree.

Do we have that $e_b$ and $f_b$ are independent of $b$?

I'm thinking about the topological point of view. When you have a topological covering $X\to X/G$ obtained by a free action of a finite group on a compact connected Riemann surface, you have that all ramification indices of points lying over a fixed point are equal.

Yes, basically by Sun-Ze's theorem (a.k.a. "the chinese remainder theorem), the Galois group is transitive on primes $b$ lying over a given prime $a$ of $A$, from which these numbers are found to be identical for all such. (Of course, lest there be confusion, as the point/prime $a$ downstairs varies, the configurations of ramification and number of points lying over will generally be different. The "generic" case has all ramification trivial, but, still, the residue field can vary. For abelian covers/extensions, classfield theory and the extension of Dirichlet's theorem on primes in arithmetic progressions prove that all possible unramified behaviors occur, with equal asymptotic densities.)