Computing automorphisms of abelian number fields
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- by Vincenzo Acciaro and Jürgen Klüners PDF
- Math. Comp. 68 (1999), 1179-1186 Request permission
Abstract:
Let $L=\mathbb {Q}(\alpha )$ be an abelian number field of degree $n$. Most algorithms for computing the lattice of subfields of $L$ require the computation of all the conjugates of $\alpha$. This is usually achieved by factoring the minimal polynomial $m_{\alpha }(x)$ of $\alpha$ over $L$. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of $\alpha$, which is based on $p$-adic techniques. Given $m_{\alpha }(x)$ and a rational prime $p$ which does not divide the discriminant $\operatorname {disc} (m_{\alpha }(x))$ of $m_{\alpha }(x)$, the algorithm computes the Frobenius automorphism of $p$ in time polynomial in the size of $p$ and in the size of $m_{\alpha }(x)$. By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of $\alpha$.References
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Additional Information
- Vincenzo Acciaro
- Affiliation: Dipartimento di Informatica, Università degli Studi di Bari, via E. Orabona 4, Bari 70125, Italy
- Email: acciaro@di.uniba.it
- Jürgen Klüners
- Affiliation: Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
- ORCID: 0000-0001-6825-307X
- Email: klueners@iwr.uni-heidelberg.de
- Received by editor(s): December 6, 1995
- Received by editor(s) in revised form: July 29, 1996
- Published electronically: February 8, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 1179-1186
- MSC (1991): Primary 11R37; Secondary 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-99-01084-4
- MathSciNet review: 1648426