The Monogeneity of Kummer Extensions and Radical Extensions

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.