The Monogeneity of Kummer Extensions and Radical Extensions

TL;DR

This paper characterizes when Kummer and radical extensions are monogenic, providing necessary and sufficient conditions, and explores the relationship between ramification and monogeneity in these extensions.

Contribution

It generalizes criteria for monogeneity in Kummer and radical extensions over arbitrary number fields, including conditions involving ramification.

Findings

Criteria for monogeneity of Kummer extensions over Q

Conditions for radical extensions to have power basis

Relation between ramification and monogeneity

Abstract

We give necessary and sufficient conditions for the Kummer extension $K:=\mathbb{Q}\left(\zeta_n,\sqrt[n]{\alpha}\right)$ to be monogenic over $\mathbb{Q}(\zeta_n)$ with $\sqrt[n]{\alpha}$ as a generator, i.e., for $\mathcal{O}_K=\mathbb{Z}\left[\zeta_n\right]\left[\sqrt[n]{\alpha}\right]$. We generalize these ideas to radical extensions of an arbitrary number field $L$ and provide necessary and sufficient conditions for $\sqrt[n]{\alpha}$ to generate a power $\mathcal{O}_L$-basis for $\mathcal{O}_{L\left(\sqrt[n]{\alpha}\right)}$. We also give sufficient conditions for $K$ to be non-monogenic over $\mathbb{Q}$ and establish a general criterion relating ramification and relative monogeneity. Using this criterion, we find a necessary and sufficient condition for a relative cyclotomic extension of degree $\phi(n)$ to have $\zeta_n$ as a monogenic generator.