Computation of the Splitting Field of a Dihedral Polynomial

Guénaël Renault (1)

1 SPIRAL - Systèmes Polynomiaux, Implantation, Résolution Algébrique

Let g be a univariate separable polynomial of degree n with coefficients in a computable field K and let (α1, . . . , αn) be an n-tuple of its roots in an algebraic closure K ̄ of K. Obtaining an algebraic representation of the splitting field K(α1,...,αn) of g is a question of first importance in effective Galois theory. For instance, it allows to manipulate symbolically the roots of g. In this paper, we focus on the computation of the splitting field of g when its Galois group is a dihedral group. We provide an algorithm for this task which returns a triangular set encoding the relations ideal of g which has a degree 2n since the Galois group of g is dihedral. Our algorithm starts from a factorization of g in K[X]/⟨g⟩ and constructs the searched triangular set by performing at most n^2 computations of normal forms modulo an ideal of degree 2n.

Mots clés: Galois Theory; Splitting Field; Symbolic Computation

Domaines: Calcul formel [cs.SC]

Guénaël Renault. Computation of the Splitting Field of a Dihedral Polynomial. International Symposium on Symbolic and Algebraic Computation, Jul 2006, Genova, Italy. pp.290-297, ⟨10.1145/1145768.1145816⟩. ⟨hal-01351454⟩