Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers in L is free as a module over the associated order . We also give examples, some of which show that this result can still hold without the assumption that K contains a primitive pth root of unity.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-2-5,
author = {James E. Carter},
title = {Some remarks on Hilbert-Speiser and Leopoldt fields of given type},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {217-223},
zbl = {1111.11051},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-2-5}
}
James E. Carter. Some remarks on Hilbert-Speiser and Leopoldt fields of given type. Colloquium Mathematicae, Tome 107 (2007) pp. 217-223. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-2-5/