A New Computational Approach to Ideal Theory in Number Fields

作者：Guardia Jordi; Montes Jesus; Nart Enric^*

来源：Foundations of Computational Mathematics, 2013, 13(5): 729-762.

DOI：10.1007/s10208-012-9137-5

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

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.

• 出版日期2013-10