A New Computational Approach to Ideal Theory in Number Fields |
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Authors: | Jordi Guàrdia Jesús Montes Enric Nart |
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Institution: | 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 2. Departament de Ciències Econòmiques i Empresarials, Facultat de Ciències Socials, Universitat Abat Oliba CEU, Bellesguard 30, 08022, Barcelona, Catalonia, Spain 3. Departament de Matemàtica Econòmica, Financera i Actuarial, Facultat d’Economia i Empresa, Univ. de Barcelona, Av. Diagonal 690, 08034, Barcelona, Catalonia, Spain 4. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193, Bellaterra, Barcelona, Catalonia, Spain
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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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