Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

Let $L$ be a number field. We give necessary and sufficient conditions for a radical extension $L(\!\sqrt [n]{\alpha })$ to be monogenic over $L$ with $\sqrt [n]{\alpha }$ as a generator, i.e., for $\sqrt [n]{\alpha }$ to generate a power $\mathcal {O}_L$-basis for the ring of integers $\mathcal {O}_{L(\!\sqrt [n]{\alpha })}$. We also give sufficient conditions for a Kummer extension of the form $\mathbb {Q}(\zeta _n,\sqrt [n]{\alpha })$ 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.