Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On integral bases in relative quadratic extensions

Authors: M. Daberkow and M. Pohst
Journal: Math. Comp. 65 (1996), 319-329
MSC (1991): Primary 11R04, 11R20, 11Y40
MathSciNet review: 1325866
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal F$ be an algebraic number field and $\mathcal E$ a quadratic extension with $\mathcal E=\mathcal F(\sqrt {\mu})$. We describe a minimal set of elements for generating the integral elements $o_{\mathcal E}$ of $\mathcal E$ as an $o_{\mathcal F}$ module. A consequence of this theoretical result is an algorithm for constructing such a set. The construction yields a simple procedure for computing an integral basis of $\mathcal E$ as well. In the last section, we present examples of relative integral bases which were computed with the new algorithm and also give some running times.

References [Enhancements On Off] (What's this?)

  • 1 E. Artin, Questions de base minimale dans la th$\acute{e}$orie des nombres alg$\acute{e}$briques, The collected papers of Emil Artin, Addison--Wesley, Reading, MA, 1965, pp. 229--231. MR 31:1159
  • 2 J.W.S. Cassels, Local fields, Cambridge Univ. Press, Cambridge, 1986. MR 87i:11172
  • 3 Fachgruppe Computeralgebra der GI, Computeralgebra in Deutschland, Fachgruppe Computeralgebra der GI (1993), 212 -- 218.
  • 4 U. Fincke and M. Pohst, A procedure for determining algebraic integers of given norm, Proc. Eurosam 83, Springer Lecture Notes in Comput. Sci., vol. 162, 1983, pp. 194 -- 202. MR 86k:11078
  • 5 A. Fröhlich, Discriminants of algebraic number fields, Math. Z. 74 (1960), 18 -- 28. MR 22:4707
  • 6 H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Jahresber. Deutsch. Math.-Verein. 35 (1926).
  • 7 E. Hecke, Lectures on the theory of algebraic numbers, Springer-Verlag, New York, 1981. MR 83m:12001
  • 8 D. Hilbert, Über die Theorie des relativquadratischen Zahlkörpers, Math. Ann. 51 (1898).
  • 9 W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer-Verlag, New York, 1990. MR 91h:11107
  • 10 J. Sommer, Vorlesungen über Zahlentheorie, Teubner, Leipzig, 1907.
  • 11 H. Zassenhaus, Ein Algorithmus zur Berechnung einer Minimalbasis über gegebener Ordnung, Funktionalanalysis, Approximationstheorie, Numerische Mathematik (Oberwolfach, 1965), Birkhäuser, Basel, 1967, pp. 90--103. MR 37:2720

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 11R04, 11R20, 11Y40

Retrieve articles in all journals with MSC (1991): 11R04, 11R20, 11Y40

Additional Information

M. Daberkow
Affiliation: Technische Universität Berlin, Fachbereich 3, Sekr. Ma8-1, Straße des 17. Juni 136, 10623 Berlin, Germany

M. Pohst
Affiliation: Technische Universität Berlin, Fachbereich 3, Sekr. Ma8-1, Straße des 17. Juni 136, 10623 Berlin, Germany

Received by editor(s): June 17, 1994
Received by editor(s) in revised form: November 29, 1994
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society