Integers for Radical Extensions of Odd Prime Degree as Product of Subrings

TL;DR

This paper investigates the structure of the ring of integers in radical extensions of odd prime degree, showing it can be decomposed into subrings with specific properties related to prime divisors of the discriminant.

Contribution

It introduces a novel decomposition of the ring of integers into subrings with prime divisor properties and provides criteria for monogeneity in such extensions.

Findings

Ring of integers decomposes into subrings with q-maximal factors

Discriminant of the ring is the gcd of factors' discriminants

Criterion for monogeneity of radical extensions established

Abstract

For a radical extension K of odd prime degree the ring O_K of integers is constructed as a product of subrings with the following property: for all prime divisors q of the discriminant of O_K there is a q-maximal factor. The discriminant of O_K is the greatest common divisor of the discriminants of all factors. The results are applied to give a criterion for the monogeneity of K where the opposite is not true.