A Dedekind's Criterion over Valued Fields

TL;DR

This paper generalizes Dedekind's criterion to arbitrary-rank valued fields, providing necessary and sufficient conditions for the integral closedness of extensions, with applications and examples illustrating the theory.

Contribution

It extends Dedekind's criterion to valued fields of arbitrary rank, offering a comprehensive characterization of integral closedness in this broader context.

Findings

Characterization of integral closedness of $R_ u[ heta]$

Necessary and sufficient conditions based on valuation extensions

Applications demonstrating the theoretical results

Abstract

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient conditions for $R_\nu[\alpha]$ to be integrally closed. We further characterize the integral closedness of $R_\nu[\alpha]$ based on information about the valuations on $K(\alpha)$ extending $\nu$. Our results enhance and generalize some existing results in the relevant literature. Some applications and examples are also given.