A New Computational Approach to Ideal Theory in Number Fields

Jordi Guàrdia, Jesús Montes, Enric Nart

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12 Citations (Scopus)


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. © 2012 SFoCM.
Original languageEnglish
Pages (from-to)729-762
JournalFoundations of Computational Mathematics
Issue number5
Publication statusPublished - 1 Oct 2013


  • Discriminant
  • Fractional ideal
  • Montes algorithm
  • Newton polygon
  • Number field
  • p-adic factorization

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