Dedekind's criterion for the monogenicity of a number field versus Uchida's and L\"uneburg's

TL;DR

This paper compares three criteria for determining when a number field's ring of integers is generated by a single element, and applies these results to specific quadratic field towers.

Contribution

It provides a comparative analysis of Dedekind's, Uchida's, and L"uneburg's characterizations of monogenicity in number fields, with applications to concrete examples.

Findings

Dedekind's criterion aligns with Uchida's and L"uneburg's in specific cases

New insights into the monogenicity of quadratic field towers

Explicit criteria for monogenicity in selected number field examples

Abstract

We compare three different characterizations, due respectively to R. Dedekind, K. Uchida, and H. L\"uneburg, of when $\mathbb Z[\theta]$ is the ring of integers of $\mathbb Q(\theta)$, and apply our results to some concrete $2$-towers of number fields.