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.References
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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
- Email: ferreirp@matesco.unican.es
- M. Angeles Gomez-Molleda
- Affiliation: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39005 Santander, Spain
- Email: gomezma@matesco.unican.es
- 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)
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 2043-2060
- MSC (2000): Primary 12Y05; Secondary 68W30, and, 11R32
- DOI: https://doi.org/10.1090/S0025-5718-03-01620-X
- MathSciNet review: 2059750