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1995 | 70 | 3 | 279-286
Tytuł artykułu

On relative integral bases for unramified extensions

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EN
Abstrakty
EN
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].
Słowa kluczowe
Czasopismo
Rocznik
Tom
70
Numer
3
Strony
279-286
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-07-06
Twórcy
  • Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland
Bibliografia
  • [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.
  • [4] L. McCulloh, Frobenius groups and integral bases, J. Reine Angew. Math. 248 (1971), 123-126.
  • [5] E. Steinitz, Rechteckige Systeme und Moduln in algebraischen Zahlkörpern I, II, Math. Ann. 71 (1911), 328-353; 72 (1911), 297-345.
  • [6] K. Uchida, Unramified extensions of quadratic number fields I, Tôhoku Math. J. 22 (1970), 138-141
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-aav70i3p279bwm
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