On the Integral Closedness of $R[\alpha]$

TL;DR

This paper presents a simplified computational criterion for the integral closedness of rings generated by algebraic elements over Dedekind rings, with specific conditions for rings of integers in number fields.

Contribution

It introduces an easier version of Dedekind's criterion and provides new necessary and sufficient conditions for integral closedness of $R[ heta]$ in number fields.

Findings

Simplified Dedekind's criterion for computational use

Necessary and sufficient conditions for $R[ heta]$ to be integrally closed

Examples illustrating the criteria and conditions

Abstract

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves those versions know in the literature. We further use this result to give a sufficient condition for the integral closedness of $R[\alpha]$ when $f(x)=x^n-a$. In case $R$ is a ring of integers of a number field, we give yet sufficient and necessary conditions for this to hold, generalizing and improving in both cases some known results in this direction. Some highlighting examples are also given.