In this paper 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 (Thm. 11). 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 Q(ζm , m√a1 , . . . , m√an )/Q(ζm ), with ai ∈ Z. We prove that these extensions always have trivial Steinitz classes. We also give sufficient condition for the existence of a normal integral basis for such extensions and an example showing that such conditions are sharp in the general case. An accurate study of the ramification produces explicit necessary and sufficient conditions on the elements ai for the extension to be tame.
Normal integral bases and tameness conditions for Kummer extensions
DEL CORSO, ILARIA;
2013-01-01
Abstract
In this paper 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 (Thm. 11). 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 Q(ζm , m√a1 , . . . , m√an )/Q(ζm ), with ai ∈ Z. We prove that these extensions always have trivial Steinitz classes. We also give sufficient condition for the existence of a normal integral basis for such extensions and an example showing that such conditions are sharp in the general case. An accurate study of the ramification produces explicit necessary and sufficient conditions on the elements ai for the extension to be tame.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
