On the Equality $\sum_{j} e_jf_j=[L:K]$ \vspace{2mm} On the Equality $\sum_{j} e_jf_j=[L:K]$ for a Finite Separable Extension $L$ of $K$ (2412.19224v2)

Abstract: Let $v$ be a discrete valuation of a field $K$, which indicates that the valuation group of $v$ is isomorphic to the integers $\mathbb{Z}$ with the natural order, and let $L$ be a finite separable extension of $K$ with a complete set ${V_1,V_2,...,V_g}$ of extended valuations of $v$. Then it is well-known that the following basic equation holds: [\sum_{j=1}^{g} e_jf_j= [L:K],] where $e_j$ and $f_j$ denote the ramification index and the relative degree for each $j$, respectively. We extend this result to the case when $v$ is a semi-discrete valuation, indicating that the valuation group is isomorphic to $\mathbb{Z}^n\ (n\geq 1)$ with lexicographic order. As a corollary to this result, we show that it is necessary and sufficient for the integral closure $D$ of the valuation ring $A$ of $v$ to be a free $A$-module that all prime ideals of $D$ other than the maximal ideals are unramified.