On a Theorem of Dedekind

TL;DR

This paper extends Dedekind's criterion to arbitrary valued fields, characterizing when the polynomial ring is integrally closed without assuming separability, and proves the theorem and its converse in this general setting.

Contribution

It generalizes Dedekind's criterion to non-separable extensions over arbitrary valued fields, providing a complete characterization of integral closedness.

Findings

Dedekind's theorem holds without separability assumption.

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

Extension of Dedekind's criterion to arbitrary valued fields.

Abstract

Let $(K,\nu)$ be an arbitrary valued field with valuation ring $R_{\nu}$ and $L=K(\alpha)$, where $\alpha$ is a root of a monic irreducible polynomial $f\in R_{\nu}[x]$. In this paper, we characterize the integral closedness of $R_\nu[\alpha]$ in such a way that extend Dedekind's criterion. Without the assumption of separability of the extension $L/K$, we show that Dedekind's theorem and its converse hold.