On the Equality ${\sum }_j{e}_j{f}_j=[L:K]$ \vspace{2mm} On the Equality ${\sum }_j{e}_j{f}_j=[L:K]$ for a Finite Separable Extension $L$ of $K$

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 ${\mathbb{Z}}^n(n\ge 1)$ 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.

I was informed that result in my paper contains contaradiction, so that I want to study the situation more deeply