The Monogeneity of Kummer Extensions and Radical Extensions

Abstract

We give necessary and sufficient conditions for the Kummer extension Q (ζn, n √ α) to be monogenic over Q(ζn) with n √ α as a generator, i.e., forWe generalize these ideas to radical extensions of an arbitrary number field L and provide necessary and sufficient conditions for n √ α to generate a power OL-basis for O L( n √ α) . We also give sufficient conditions for K to be non-monogenic over Q and establish some general criteria relating ramification and relative monogeneity.