On relative integral bases for unramified extensions

Kevin Hutchinson. "On relative integral bases for unramified extensions." Acta Arithmetica 70.3 (1995): 279-286. <http://eudml.org/doc/206751>.

@article{KevinHutchinson1995,
abstract = {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].},
author = {Kevin Hutchinson},
journal = {Acta Arithmetica},
keywords = {Dedekind domain; separable extension; fractional ideals; unramified extension of odd degree; relative integral basis},
language = {eng},
number = {3},
pages = {279-286},
title = {On relative integral bases for unramified extensions},
url = {http://eudml.org/doc/206751},
volume = {70},
year = {1995},
}

TY - JOUR
AU - Kevin Hutchinson
TI - On relative integral bases for unramified extensions
JO - Acta Arithmetica
PY - 1995
VL - 70
IS - 3
SP - 279
EP - 286
AB - 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].
LA - eng
KW - Dedekind domain; separable extension; fractional ideals; unramified extension of odd degree; relative integral basis
UR - http://eudml.org/doc/206751
ER -