Computing the Galois group of a polynomial using linear differential equations

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Computing the Galois group of a polynomial using linear differential equations

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International Conference on Symbolic and Algebraic Computation archive
Proceedings of the 2000 international symposium on Symbolic and algebraic computation table of contents

St. Andrews, Scotland

Pages: 78 - 85

Year of Publication: 2000

ISBN:1-58113-218-2

Authors

Olivier Cormier	IRMAR, Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex
Michael F. Singer	Dept. of Math., Box 8205, NC State University, Raleigh, NC
Felix Ulmer	IRMAR, Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex

Sponsor

SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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ACM New York, NY, USA

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ABSTRACT

In this paper we show how to compute the Galois group G of a polynomial ƒ ∈ Q(x)[Y] by factoring the associated linear differential equation Lƒ(Y) = 0 (and constructions of it) of minimal order satisfied by the roots of ƒ. We use that the differential Galois group of Lƒ(Y) is a faithful linear representation of G whose character is a summand of the permutation character of G acting on the roots of ƒ. Our approach is motivated by the fact that the orders of the involved differential equations are much lower than the degrees of the Lagrange resolvants of ƒ. In the final section we show how, if ƒ ∈ Q(x)[Y], our approach via differential Galois theory helps one to also compute the Galois group of ƒ over Q(x).

REFERENCES

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