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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 the defining equation $f(x)$. 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.
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GUÀRDIA, Jordi, MONTES, Jesús and NART, Enric. A new computational approach to ideal theory in number fields. Foundations of Computational Mathematics. 2013. Vol. 13, num. 5, pags. 729-762. ISSN 1615-3375. [consulted: 22 of October of 2026]. Available at: https://hdl.handle.net/2445/124824