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Computation of Galois groups of rational polynomials

Part of: Algebraic number theory: global fields Computational number theory Ring extensions and related topics

Published online by Cambridge University Press:  01 May 2014

Claus Fieker  and
Jürgen Klüners
Claus Fieker
Affiliation:
Faculty of Mathematics,  University of Kaiserlautern,  P.O. Box 3049,  D-67653 Kaiserslautern,  Germany email fieker@mathematik.uni-kl.de
Jürgen Klüners
Affiliation:
Mathematisches Institut der Universität Paderborn,  Warburger Str. 100,  D-33098 Paderborn,  Germany email klueners@math.uni-paderborn.de

Abstract

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Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups $H<G$, a $G$-relative $H$-invariant, that is a multivariate polynomial $F$ that is $H$-invariant, but not $G$-invariant. While generic, theoretical methods are known to find such $F$, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.

MSC classification

Secondary: 11R32: Galois theory 13B05: Galois theory 11Y40: Algebraic number theory computations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 141 - 158
Copyright
© The Author(s) 2014 

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