A Modular Method for Computing the Splitting Field of a Polynomial

Abstract : 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.
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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⟩



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