A new computational approach to ideal theory in number fields
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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 Montes algorithm, which is an extremely fast procedure to construct the prime ideals
CitationGuàrdia, J.; Montes, J.; Nart, E. A new computational approach to ideal theory in number fields. "Foundations of computational mathematics", 2013, vol. 13, núm. 5, p. 729-762.