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

Fernandez-Ferreiros, Pilar; Gomez-Molleda, M.

doi:10.1090/S0025-5718-03-01620-X

Deciding the nilpotency of the Galois group by computing elements in the centre
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by Pilar Fernandez-Ferreiros and M. Angeles Gomez-Molleda PDF

Math. Comp. 73 (2004), 2043-2060 Request permission

Abstract:

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.

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