On a Theorem of Dedekind

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.

8 pages (To appear in SJM)