Mathematics of Computation

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Computing class fields via the Artin map

Author(s): Claus Fieker.
Journal: Math. Comp. 70 (2001), 1293-1303.
MSC (2000): Primary 11Y40; Secondary 11R37
Posted: March 24, 2000
MathSciNet review: 1826583
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

Based on an explicit representation of the Artin map for Kummer extensions, we present a method to compute arbitrary class fields. As in the proofs of the existence theorem, the problem is first reduced to the case where the field contains sufficiently many roots of unity. Using Kummer theory and an explicit version of the Artin reciprocity law we show how to compute class fields in this case. We conclude with several examples.

References:

1.: V. Acciaro and J. Klüners.
Computing automorphisms of abelian number fields.
Math. Comp., 68(227):1179-1186, 1999. MR 99i:11099
2.: E. Bach and J. Sorenson.
Explicit bounds for primes in residue classes.
Math. Comp., 65(216):1717 - 1735, 1996. MR 97a:11143
3.: H. Bauer.
Zur Berechnung von Hilbertschen Klassenkörpern mit Hilfe von Stark-Einheiten.
Diploma thesis, Technische Universität Berlin, 1998.
4.: H. Cohen.
A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics.
Springer, 1 edition, 1993. MR 94i:11105
5.: H. Cohen.
Advanced topics in computational number theory, Volume 193 of Graduate Texts in Mathematics. Springer, 1 edition, 1999. CMP 2000:05
6.: H. Cohen, F. Diaz y Diaz, and M. Olivier.
Computing ray class groups, conductors and discriminants.
Math. Comp., 67(222):773-795, 1998. MR 98g:11128
7.: M. Daberkow and M. E. Pohst.
On the computation of Hilbert class fields.
J. Number Theory, 69:213-230, 1998. MR 99h:11130
8.: D. S. Dummit, J. W. Sands, and B. A. Tangedal.
Computing Stark units for totally real cubic fields.
Math. Comp., 66(219):1239-1267, 1997. MR 97i:11110
9.: C. Fieker.
Über relative Normgleichungen in algebraischen Zahlkörpern.
PhD thesis, Technische Universität Berlin, 1997.
10.: H. Hasse.
Vorlesungen über Klassenkörpertheorie.
Physika Verlag, Würzburg, 1967. MR 36:3752
11.: G. J. Janusz.
Algebraic number fields, volume 7 of Graduate studies in Mathematics.
AMS, 1996. MR 96j:11137
12.: KANT Group.
KANT V4.
J. Symb. Comp., 24:267-283, 1997.
13.: J. Klüners.
Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper.
PhD thesis, Technische Universität Berlin, 1997.
14.: H. Koch.
Number Theory II, volume 62 of Encyclopaedia of Mathematical Sciences.
Springer, Berlin, 1992.
15.: S. Lang.
Algebraic Number Theory, volume 110 of Graduate Texts in Mathematics.
Springer, 2nd edition, 1994. MR 95f:11085
16.: J. Neukirch.
Algebraische Zahlentheorie.
Springer, 1992. MR 92a:01057
17.: S. Pauli.
Zur Berechnung von Strahlklassengruppen.
Diploma thesis, Technische Universität Berlin, 1996.
18.: X.-F. Roblot.
Algorithmes de factorisation dans les corps de nombres et applications de la conjecture de Stark à la construction des corps de classes de rayon.
PhD thesis, Université Bordeaux I, 1997.
19.: R. Schertz.
Zur expliziten Berechnung von Ganzheitsbasen in Strahlklassenkörpern über imaginär-quadratischen Zahlkörpern.
J. Number Theory, 34:41-53, 1990. MR 91e:11134
20.: I. R. Shafarevich.
A new proof of the Kronecker-Weber theorem.
In Collected mathematical papers, pages 54-58. Springer, 1989.

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