Multi-modular Algorithm for Computing the Splitting Field of a Polynomial

International audience; Let f be a univariate monic integral polynomial of degree n and let (α1, ..., αn) be an n-tuple of its roots in an algebraic closure Q of Q. Obtaining an algebraic representation of the splitting field Q(α1, ..., αn) of f is a question of first importance in effective Galois theory. For instance, it allows us to manipulate symbolically the roots of f. In this paper, we propose a new method based on multi-modular strategy. Actually, we provide algorithms for this task which return a triangular set encoding the splitting ideal of f. We examine the ability/practicality of the method by experiments on a real computer and study its complexity.