Computation of the splitting field of a dihedral polynomial

Let g be a univariate separable polynomial of degree n with coefficients in a computable field double-struck K sign and let (α₁..., α_n) be an n-tuple of its roots in an algebraic closure K of K. Obtaining an algebraic representation of the splitting field double-struck K sign(α₁,..., α_n) 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 double-struck K sign[X]/〈g〉 and constructs the searched triangular set by performing n² computations of normal forms modulo an ideal of degree 2n.