Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T21:09:52.287Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Computation of Integral Closures of Cyclic Extensions of Function Fields

Published online by Cambridge University Press:  01 February 2010

Robert Fraatz
Robert Fraatz
Affiliation:
Technische Universität Berlin, Fakultät II, Institut für Mathematik MA 8–1, Straße des 17. Juni 136, 10623 Berlin, Germany, fraatz@math.tu-berlin.de, http://www.math.tu-berlin.de/~fraatz

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let S be a non-empty proper subset of the set of places of a global function field F and E a cyclic Kummer or Artin–Schreier–Witt extension of F. We present a method of efficiently computing the ring of elements of E which are integral at all places of S. As an important tool, we include an algorithmic version of the strong approximation theorem. We conclude with several examples.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 141 - 160
Copyright
Copyright © London Mathematical Society 2007

References

1.Buhler, J. P., Lenstra, H. W., Jr and Pomerance, Carl, ‘Factoring integers with the number field sieve’, The development of the number field sieve, Lecture Notes in Math. 1554 (Springer, Berlin, 1993) 50–94.CrossRefGoogle Scholar
2.Cannon, J. et al. , The computer algebra system MAGMA (University of Syd ney, 2004), http://magma.maths.usyd.edu.au/magma/.Google Scholar
3.Cassels, J.W.S., Local fields, London Mathematical Society Student Texts 3 (Cambridge University Press, Cambridge, 1986).Google Scholar
4.Fraatz, R., ‘Computation of maximal orders of cyclic extensions of function fields’, thesis, Technische Universität Berlin, (2005), http://www.math.tu-berlin.de/~kant/publications/diss/diss_Robert-Fraatz.pdf.Google Scholar
5.Friedrichs, C., ‘Bestimmung relativer Ganzheitsbasen mit dem Round-2-Algorithmus’, Diplomarbeit, Technische Universität Berlin, (1997), http://www.math.tu-berlin.de/~kant/publications/diplom/friedrichs.ps.gzGoogle Scholar
6.Friedrichs, C., ‘Berechnung von Maximalordnungen über Dedekindringen’, thesis, Technische Universität Berlin, (2000), http://www.math.tu-berlin.de/~kant/publications/diss/diss_fried.pdfGoogle Scholar
7.Goppa, V. D., ‘Codes on algebraic curves’, ,Dokl. Akad. Nauk SSSR 259 (1981) 1289–1290.Google Scholar
8.Goppa, V. D., Geometry and codes, Mathematics and its Applications (Soviet Series) 24 (Kluwer, Dordrecht, 1988).Google Scholar
9.Hasse, H., ‘Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper’, J. Reine Angew. Math. 172 (1934) 37–54.Google Scholar
10.Hess, F., ‘Computing Riemann-Roch spaces in algebraic function fields and related topics’, J. Symbolic Comput. 33 (2002) 425–445.CrossRefGoogle Scholar
11.Lorenz, F., Einführung in die Algebra. Teil II (Bibliographisches Institut, Mannheim, 1990).Google Scholar
12.Pohst, M. and Zassenhaus, H., Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications 30 (Cambridge University Press, Cambridge, 1997).Google Scholar
13.Schmid, H. L., ‘Zyklische algebraische Funktionenkörper vom Grade pn über endlichem Konstantenkörper der Charakteristik p’ J. Reine Angew. Math. 175 (1936) 108–123.CrossRefGoogle Scholar
14.Stichtenoth, H., Algebraic function fields and codes, Universitext (Springer, Berlin, 1993).Google Scholar