### 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. © 2012 SFoCM.

Original language | English |
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Pages (from-to) | 729-762 |

Journal | Foundations of Computational Mathematics |

Volume | 13 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Oct 2013 |

### Keywords

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

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## Cite this

Guàrdia, J., Montes, J., & Nart, E. (2013). A New Computational Approach to Ideal Theory in Number Fields.

*Foundations of Computational Mathematics*,*13*(5), 729-762. https://doi.org/10.1007/s10208-012-9137-5