A new  approach to  Computational Algebraic Number Theory: The '+Ideals' package
J. Guàrdia, E. Nart

Abstract

We will present the +Ideals package for Magma, designed to compute in number fields. The cornerstone in Computational Algebraic Number Theory is the computation of integral bases, which is a  hard problem in many real situations. The idea behind our package is to skip this problem by using a new representation of the prime ideals.
This representation leads to  completely different algorithms to carry out the common tasks in Algebraic Number Theory. The key ingredient of these procedures is Montes algorithm, based on  Newton polygons of higher order. The new algorithms turn to be extremely efficient and they expand  the actual computational limits by far.

In the talk we will briefly present this new philosophy and illustrate the use of the package with some computations in big number fields. The last part of the talk will be devoted to discuss the potential application of the package in problems suggested by the audience.