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A New Computational Approach to Ideal Theory in Number Fields

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Abstract

Let K be the number field determined by a monic irreducible polynomial f(x) with integer coefficients. In previous papers we parameterized the prime ideals of K in terms of certain invariants attached to Newton polygons of higher order of f(x). In this paper we show how to carry out the basic operations on fractional ideals of K in terms of these constructive representations of the prime ideals. From a computational perspective, these results facilitate the manipulation of fractional ideals of K avoiding two heavy tasks: the construction of the maximal order of K and the factorization of the discriminant of f(x). The main computational ingredient is the Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

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Acknowledgements

Research partially supported by MTM2009-13060-C02-02 and MTM2009-10359 from the Spanish MEC

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada IV, Escola Politècnica Superior d’Enginyera de Vilanova i la Geltrú, Av. Víctor Balaguer s/n., 08800, Vilanova i la Geltrú, Catalonia, Spain

    Jordi Guàrdia

  2. Departament de Ciències Econòmiques i Empresarials, Facultat de Ciències Socials, Universitat Abat Oliba CEU, Bellesguard 30, 08022, Barcelona, Catalonia, Spain

    Jesús Montes

  3. Departament de Matemàtica Econòmica, Financera i Actuarial, Facultat d’Economia i Empresa, Univ. de Barcelona, Av. Diagonal 690, 08034, Barcelona, Catalonia, Spain

    Jesús Montes

  4. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Enric Nart

Authors
  1. Jordi Guàrdia
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  2. Jesús Montes
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  3. Enric Nart
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Corresponding author

Correspondence to Enric Nart.

Additional information

Communicated by Felipe Cucker.

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Guàrdia, J., Montes, J. & Nart, E. A New Computational Approach to Ideal Theory in Number Fields. Found Comput Math 13, 729–762 (2013). https://doi.org/10.1007/s10208-012-9137-5

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  • DOI: https://doi.org/10.1007/s10208-012-9137-5

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