0. Introduction. Since ℤ is a principal ideal domain, every finitely generated torsion-free ℤ-module has a finite ℤ-basis; in particular, any fractional ideal in a number field has an "integral basis". However, if K is an arbitrary number field the ring of integers, A, of K is a Dedekind domain but not necessarily a principal ideal domain. If L/K is a finite extension of number fields, then the fractional ideals of L are finitely generated and torsion-free (or, equivalently, finitely generated and projective) as A-modules, but not necessarily free. Beginning with some classical results of Artin and Chevalley (Propositions 1.1 and 1.2), we give some criteria for the existence or nonexistence of A-bases for ideals in L or for the ring of integers of L in the case where L/K is unramified (Theorem 1.10 and Corollary 2.3). In particular, we show how the existence of an integral basis is (under mild hypotheses) determined by purely group-theoretic properties of the Galois group of the normal closure of L/K. We prove the main results for arbitrary finite separable field extensions L/K. The arguments were suggested by reading [4].
@article{bwmeta1.element.bwnjournal-article-aav70i3p279bwm,
author = {Kevin Hutchinson},
title = {On relative integral bases for unramified extensions},
journal = {Acta Arithmetica},
volume = {69},
year = {1995},
pages = {279-286},
zbl = {0815.11054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav70i3p279bwm}
}
Kevin Hutchinson. On relative integral bases for unramified extensions. Acta Arithmetica, Tome 69 (1995) pp. 279-286. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav70i3p279bwm/
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