A Dedekind's Criterion over Valued Fields

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.

10 pages