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

Authors: Pilar Fernandez-Ferreiros and M. Angeles Gomez-Molleda
Journal: Math. Comp. 73 (2004), 2043-2060
MSC (2000): Primary 12Y05; Secondary 68W30, and, 11R32
Published electronically: November 3, 2003
MathSciNet review: 2059750
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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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Additional Information

Pilar Fernandez-Ferreiros
Affiliation: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39005 Santander, Spain

M. Angeles Gomez-Molleda
Affiliation: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39005 Santander, Spain

Received by editor(s): May 24, 2002
Received by editor(s) in revised form: March 16, 2003
Published electronically: November 3, 2003
Additional Notes: Partially supported by the grant DGESIC PB 98-0713-C02-02 (Ministerio de Educacion y Cultura)
Article copyright: © Copyright 2003 American Mathematical Society

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