| maths.univ-rennes1.fr (2007) | |||||||||||||||||
Abstract | |||||||||||||||||
| maths.univ-rennes1.fr In this paper we show how to compute the Galois group G of a polynomial f 2 Q(x)[Y] by factoring the associated linear differential equation Lf (Y) = 0 (and constructions of it) of minimal order satisfied by the roots of f. We use that the differential Galois group of Lf (Y) is a faithful linear representation of G whose character is a summand of the permutation character of G acting on the roots of f. 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 f. In the final section we show how, if f 2 Q(x)[Y], our approach via differential Galois theory helps one to also compute the Galois group of f over Q(x). | |||||||||||||||||
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