×

An efficient algorithm for the computation of Galois automorphisms. (English) Zbl 1094.11045

Summary: We describe an algorithm for computing the Galois automorphisms of a Galois extension which generalizes the algorithm of Acciaro and Klüners to the non-Abelian case. This is much faster in practice than algorithms based on LLL or factorization.

MSC:

11Y40 Algebraic number theory computations
11R32 Galois theory
12F10 Separable extensions, Galois theory

Software:

KANT/KASH; PARI/GP; Magma
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] John ABBOTT, Victor SHOUP and Paul ZIMMERMAN, Factorization in \(\mathbb{Z} [x]\): The Searching Phase , Proc. ISSAC 2000, ACM Press, 2000, pp. 1-7. http://www.shoup.net/papers/asz.ps.Z. · Zbl 1326.68339
[2] Vincenzo Acciaro and Jürgen Klüners, Computing automorphisms of abelian number fields, Math. Comp. 68 (1999), no. 227, 1179 – 1186. · Zbl 0937.11062
[3] List of polynomials of the benchmark, http://www.math.u-bordeaux.fr/ allomber/ nfgaloisconj_benchmark.html
[4] Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. · Zbl 0786.11071
[5] Marshall Hall Jr., The theory of groups, The Macmillan Co., New York, N.Y., 1959.
[6] Jürgen KLÜNERS, Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper, Thesis, Technischen Universitat Berlin, 1997. · Zbl 0912.11059
[7] Jürgen Klüners, On computing subfields. A detailed description of the algorithm, J. Théor. Nombres Bordeaux 10 (1998), no. 2, 243 – 271 (English, with English and French summaries). · Zbl 0935.11047
[8] Jürgen Klüners and Gunter Malle, Explicit Galois realization of transitive groups of degree up to 15, J. Symbolic Comput. 30 (2000), no. 6, 675 – 716. Algorithmic methods in Galois theory. · Zbl 0967.12004
[9] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267 – 283. Computational algebra and number theory (London, 1993). · Zbl 0886.11070
[10] W. BOSMA, J. CANNON and C. PLAYOUST, The Magma algebra system I: The user language, J. Symb. Comput., 24, 1997, 235-265. CMP 98:05 · Zbl 0898.68039
[11] PARI, C. BATUT, K. BELABAS, D. BERNARDI, H. COHEN and M. OLIVIER, User’s Guide to PARI-GP, version 2.2.1.
[12] Xavier ROBLOT, Algorithmes de factorisation dans les extensions relatives et applications de la conjecture de Stark à la construction de corps de classes de rayon, Thesis, Université Bordeaux I, 1997.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.