# On relative integral bases for unramified extensions

Acta Arithmetica (1995)

- Volume: 70, Issue: 3, page 279-286
- ISSN: 0065-1036

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topKevin 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 -

## References

top- [1] E. Artin, Questions de base minimale dans la théorie des nombres algébriques, CNRS XXIV (Colloq. Int., Paris, 1949), 19-20.
- [2] D. A. Cox, Primes of the Form x² + ny², Wiley, 1989.
- [3] A. Fröhlich, Ideals in an extension field as modules over the algebraic integers in a finite number field, Math. Z. 74 (1960), 29-38. Zbl0098.03401
- [4] L. McCulloh, Frobenius groups and integral bases, J. Reine Angew. Math. 248 (1971), 123-126. Zbl0229.12009
- [5] E. Steinitz, Rechteckige Systeme und Moduln in algebraischen Zahlkörpern I, II, Math. Ann. 71 (1911), 328-353; 72 (1911), 297-345. Zbl42.0230.02
- [6] K. Uchida, Unramified extensions of quadratic number fields I, Tôhoku Math. J. 22 (1970), 138-141 Zbl0209.35602

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