A modular method for computing the splitting field of a polynomial

Authors:
Guénaël Renault;Kazuhiro Yokoyama
Affiliations:
LIP6-SPIRAL, Université Paris 6, Paris, France;Department of Mathematics, Rikkyo University, Tokyo, Japan
Venue:
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Year:
2006

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Abstract

We provide a modular method for computing the splitting field Kf of an integral polynomial f by suitable use of the byproduct of computation of its Galois group Gf by p-adic Stauduhar’s method. This method uses the knowledge of Gf with its action on the roots of f over a p-adic number field, and it reduces the computation of Kf 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.