Computation of the splitting field of a dihedral polynomial

  • Authors:

  • Guénaël Renault

  • Affiliations:

  • Laboratoire d'Informatique de l'Université Pierre et Marie Curie-Paris 6, Paris, France

  • Venue:
  • Proceedings of the 2006 international symposium on Symbolic and algebraic computation
  • Year:
  • 2006

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Abstract

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 us 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 degree 2n since the Galois group of g is dihedral. Our algorithm starts from a factorization of g in K[X]/ and constructs the searched triangular set by performing n2 computations of normal forms modulo an ideal of degree 2n.