On relative integral bases for unramified extensions

• Autorzy: Kevin Hutchinson
• Języki publikacji: 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].

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1995
• Tom: 70
• Numer: 3
• Strony: 279-286

Daty

wydano

1995

otrzymano

1994-07-06

Twórcy

autor

Kevin Hutchinson

• 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