Deciding the nilpotency of the Galois group by computing elements in the centre

Abstract

We present a new algorithm for computing the centre of the Galois group of a given polynomial f ∈ Q [ x ] f \in \mathbb {Q}[x] along with its action on the set of roots of f f , without previously computing the group. We show that every element in the centre is representable by a family of polynomials in Q [ x ] \mathbb {Q}[x] . For computing such polynomials, we use quadratic Newton-lifting and truncated expressions of the roots of f f over a p p -adic number field. As an application we give a method for deciding the nilpotency of the Galois group. If f f is irreducible with nilpotent Galois group, an algorithm for computing it is proposed.