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 $K$

TL;DR

This paper extends the fundamental equality relating ramification indices, residue degrees, and extension degree from discrete valuations to semi-discrete valuations with valuation groups isomorphic to ^n, providing new insights into valuation theory.

Contribution

It generalizes the classical ramification formula to semi-discrete valuations with valuation groups ^n, and characterizes when the integral closure is a free module over the valuation ring.

Findings

Extended the ramification formula to semi-discrete valuations.

Established a necessary and sufficient condition for the integral closure to be a free module.

Connected ramification properties with unramified prime ideals in the integral closure.

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.