| 3 1 (2007) | |||||||||||||||
Abstract | |||||||||||||||
| In this paper we show how to compute the geometric Galois group G Q(x) of a polynomial f 2 Q(x)[Y] by considering the associated linear differential equation L f (Y) = 0 (and constructions of it) satisfied by the roots of f. We use that the differential Galois group of L f (Y) is a faithful linear representation of G Q(x) whose character is a summand of the permutation character of G Q(x) acting on the roots of f. Our approach allows to make full use of representation theory (over a field of characteristic zero) in order to study permutation representations. 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 arithmetic Galois group GQ (x) of f over Q(x) if the later has some special properties. 1. From polynomials to linear differential equations In the following we will consider the field Q, but the results remain valid for any field of characteristic 0. Let f = Y m | |||||||||||||||
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