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

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