A Modular Method for Computing the Splitting Field of a Polynomial

Guénaël Renault (1) , Kazuhiro Yokoyama (2)

1 SPIRAL - Systèmes Polynomiaux, Implantation, Résolution Algébrique
2 Department of Mathematics [Rikkyo]

We provide a modular method for computing the splitting field $K_f$ of an integral polynomial $f$ by suitable use of the byproduct of computation of its Galois group $G_f$ by $p$-adic Stauduhar’s method. This method uses the knowledge of $G_f$ with its action on the roots of $f$ over a $p$-adic number field, and it reduces the computation of $K_f$ to solving systems of linear equations modulo some powers of $p$ and Hensel liftings. We provide a careful treatment on reducing computational difficulty. We examine the ability/practicality of the method by experiments on a real computer and study its complexity.

Mots clés: Galois Theory; Splitting Field

Domaines: Calcul formel [cs.SC]; Théorie des nombres [math.NT]

Guénaël Renault, Kazuhiro Yokoyama. A Modular Method for Computing the Splitting Field of a Polynomial. Algorithmic Number Theory Symposium, Jul 2006, Berlin, Germany. pp.124-140. ⟨hal-01337040⟩