Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Mobile Device Pairing
Green Open Access
Mathematics of Computation
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 Free Access

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 Emil Artin, The collected papers of Emil Artin, Edited by Serge Lang and John T. Tate, Addison–Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. MR 0176888 (31 #1159)
  • 2 J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. MR 861410 (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, Computer algebra (London, 1983) Lecture Notes in Comput. Sci., vol. 162, Springer, Berlin, 1983, pp. 194–202. MR 774811 (86k:11078),
  • 5 Albrecht Frölich, Discriminants of algebraic number fields, Math. Z 74 (1960), 18–28. MR 0113876 (22 #4707)
  • 6 H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Jahresber. Deutsch. Math.-Verein. 35 (1926).
  • 7 Erich Hecke, Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, vol. 77, Springer-Verlag, New York-Berlin, 1981. Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen. MR 638719 (83m:12001)
  • 8 D. Hilbert, Über die Theorie des relativquadratischen Zahlkörpers, Math. Ann. 51 (1898).
  • 9 Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer-Verlag, Berlin; PWN—Polish Scientific Publishers, Warsaw, 1990. MR 1055830 (91h:11107)
  • 10 J. Sommer, Vorlesungen über Zahlentheorie, Teubner, Leipzig, 1907.
  • 11 Hans Zassenhaus, Ein Algorithmus zur Berechnung einer Minimalbasis über gegebener Ordnung, Funktionalanalysis, Approximationstheorie, Numerische Mathematik (Oberwolfach, 1965) Birkhäuser, Basel, 1967, pp. 90–103 (German). MR 0227135 (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

PII: S 0025-5718(96)00686-2
Received by editor(s): June 17, 1994
Received by editor(s) in revised form: November 29, 1994
Article copyright: © Copyright 1996 American Mathematical Society