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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Newton polygons of higher order in algebraic number theory


Authors: Jordi Guàrdia, Jesús Montes and Enric Nart
Journal: Trans. Amer. Math. Soc. 364 (2012), 361-416
MSC (2010): Primary 11S15; Secondary 11R04, 11R29, 11Y40
Published electronically: May 18, 2011
MathSciNet review: 2833586
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Abstract: We develop a theory of arithmetic Newton polygons of higher order that provides the factorization of a separable polynomial over a $ p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by Ø. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields.


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Additional Information

Jordi Guàrdia
Affiliation: Departament de Matemàtica Aplicada IV, Escola Politècnica Superior d’Enginyera de Vilanova i la Geltrú, Av. Víctor Balaguer s/n. E-08800 Vilanova i la Geltrú, Catalonia, Spain
Email: guardia@ma4.upc.edu

Jesús Montes
Affiliation: Departament de Ciències Econòmiques i Socials, Facultat de Ciències Socials, Universitat Abat Oliba CEU, Bellesguard 30, E-08022 Barcelona, Catalonia, Spain
Email: montes3@uao.es

Enric Nart
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, E-08193 Bellaterra, Barcelona, Catalonia, Spain
Email: nart@mat.uab.cat

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05442-5
PII: S 0002-9947(2011)05442-5
Keywords: Newton polygon, local field, $p$-adic factorization, number field, prime ideal decomposition, discriminant, integral basis
Received by editor(s): October 31, 2008
Received by editor(s) in revised form: June 15, 2010
Published electronically: May 18, 2011
Additional Notes: This work was partially supported by MTM2009-13060-C02-02 and MTM2009-10359 from the Spanish MEC
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.