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Abstract

We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields $L/K$. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of $L/K$ and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer extensions $\mathbb {Q}(\zeta _m,\sqrt [m]{a_1},\dots ,\sqrt [m]{a_n})/\mathbb {Q}(\zeta _m)$ with $a_i\in \mathbb {Z}$. We prove that these extensions always have trivial Steinitz classes. We also give sufficient conditions for the existence of a normal integral basis for such extensions and an example showing that such conditions are sharp in the general case. A detailed study of the ramification produces explicit necessary and sufficient conditions on the elements $a_i$ for the extension to be tame.