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 with . 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 for the extension to be tame.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-1,
author = {Ilaria Del Corso and Lorenzo Paolo Rossi},
title = {Normal integral bases and tameness conditions for Kummer extensions},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {1-23},
zbl = {1284.11151},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-1}
}
Ilaria Del Corso; Lorenzo Paolo Rossi. Normal integral bases and tameness conditions for Kummer extensions. Acta Arithmetica, Tome 161 (2013) pp. 1-23. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-1/