# A fast algorithm for polynomial factorization over ${ℚ}_{p}$

• Volume: 14, Issue: 1, page 151-169
• ISSN: 1246-7405

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## Abstract

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We present an algorithm that returns a proper factor of a polynomial $\Phi \left(x\right)$ over the $p$-adic integers ${ℤ}_{p}$ (if $\Phi \left(x\right)$ is reducible over ${ℚ}_{p}$) or returns a power basis of the ring of integers of ${ℚ}_{p}\left[x\right]/\Phi \left(x\right){ℚ}_{p}\left[x\right]$ (if $\Phi \left(x\right)$ is irreducible over ${ℚ}_{p}$). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.

## How to cite

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Ford, David, Pauli, Sebastian, and Roblot, Xavier-François. "A fast algorithm for polynomial factorization over $\mathbb {Q}_p$." Journal de théorie des nombres de Bordeaux 14.1 (2002): 151-169. <http://eudml.org/doc/248904>.

@article{Ford2002,
abstract = {We present an algorithm that returns a proper factor of a polynomial $\Phi (x)$ over the $p$-adic integers $\mathbb \{Z\}_p$ (if $\Phi (x)$ is reducible over $\mathbb \{Q\}_p$) or returns a power basis of the ring of integers of $\mathbb \{Q\}_p[x]/ \Phi (x) \mathbb \{Q\}_p[x]$ (if $\Phi (x)$ is irreducible over $\mathbb \{Q\}_p$). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.},
author = {Ford, David, Pauli, Sebastian, Roblot, Xavier-François},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {-adic integers; polynomial factorization},
language = {eng},
number = {1},
pages = {151-169},
publisher = {Université Bordeaux I},
title = {A fast algorithm for polynomial factorization over $\mathbb \{Q\}_p$},
url = {http://eudml.org/doc/248904},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Ford, David
AU - Pauli, Sebastian
AU - Roblot, Xavier-François
TI - A fast algorithm for polynomial factorization over $\mathbb {Q}_p$
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 151
EP - 169
AB - We present an algorithm that returns a proper factor of a polynomial $\Phi (x)$ over the $p$-adic integers $\mathbb {Z}_p$ (if $\Phi (x)$ is reducible over $\mathbb {Q}_p$) or returns a power basis of the ring of integers of $\mathbb {Q}_p[x]/ \Phi (x) \mathbb {Q}_p[x]$ (if $\Phi (x)$ is irreducible over $\mathbb {Q}_p$). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.
LA - eng
KW - -adic integers; polynomial factorization
UR - http://eudml.org/doc/248904
ER -

## References

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1. [1] A. Ash, R. Pinch, R. Taylor, An Â4 extension of Q attached to a non-selfdual automorphic form on GL(3). Math. Annalen291 (1991), 753-766. Zbl0713.11036MR1135542
2. [2] G. Baier, Zum Round 4 Algorithmus, Diplomarbeit, Technische Universität Berlin, 1996, http://www.math.TU-Berlin.DE/-kant/publications/diplom/baier.ps.gz.
3. [3] H. Cohen, Personal communication, 1996.
4. [4] D. Ford, P. Letard, Implementing the Round Four maximal order algorithm. J. Théor. Nombres Bordeaux6 (1994) 39-80, http://almira.math.u-bordeaux.fr:80/jtnb/1994-1/jtnb6-1.html. Zbl0817.11064MR1305287
5. [5] E. HallouinCalcul de fermeture intégrale en dimension 1 et factorisation. Thèse, Université de Poitiers, 1998.
6. [6] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers (second edition). Springer-Verlag, Berlin, 1990. Zbl0717.11045MR1055830
7. [7] S. Pauli, Factoring Polynomials over Local Fields. Journal of Symbolic Computation, accepted 2001. Zbl1085.12500MR1858009
8. [8] X.-R. Roblot, Algorithmes de factorisation dans les extensions relatives et applications de la conjecture de Stark à la construction des corps de classes de rayon. Thèse, Université BordeauxI, 1997, http://www.desargues.univ-lyon1.fr/home/roblot/papers.htm#1.
9. [9] E. Weiss, Algebraic number theory. McGraw-Hill, 1963. Zbl0115.03601MR159805

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