Galois Theory and Reducible Polynomials

This paper presents works on reducible polynomials in Galois theory. It gives a fundamental theorem about the Galois ideals associated with a product of groups. These results are used for the practical determination of Galois groups of polynomials of degrees up to 7. : : :). The following tools are used: resolvents (relative or absolute), p-adic methods, Grr obner bases or block systems. In general, the polynomials are assumed irreducible and consequently their Galois groups are transitive (see 10], 22], 33], 38],..). The case of reducible polynomials is considered as too complicated because the Galois group of a product of polynomials is not always the product of the Galois groups of these polynomials. But, any polynomial can be studied by using partition and group matrices (see 5] and 42]). These matrices give all algorithms in order to determine the Galois group of a polynomial using factorizations of univariate polynomials called resolvents (absolute and relative). Thus, it is possible to deduce a better algorithm from these matrices. This theoretically better algorithm involves computations of relative resolvents whose degrees are small in comparison to those of absolute resolvents. In all this paper, we consider a perfect eld k and we denote by ^ k an algebraic closure of k. We will consider f a monic univariate polynomial over k of degree n. We will suppose that the polynomial f is separable (i.e. its roots are pairwise distinct). The variables x; T; x 1 ; : : : ; x n are transcendental variables over k. A resolvent of the polynomial f is a univariate polynomial; its coeecients are polynomials of kx 1 ; : : : ; x n ] invariant under the action of a subgroup M of the symmetric group of degree n which are evaluated on the roots of the polynomial f; these coeecients belong to the eld k when the group M contains the Galois group of f over k. If the group M equals the symmetric group then the resolvent is called absolute and its computation involves the fundamental theorem of symmetric functions. Otherwise, the resolvent is called M-relative. For computing relative resolvents there exist a numerical method (see 38] or 20]) and formal methods (see 3], 16] and 8]). For the numerical method, an approximation of the roots of the polynomial f is computed such that the error over the coeecients of the resolvent is strictly …