Some remarks concerning the complexity of computing class groups of quadratic fields

AbstractLet O be an order of any quadratic number field. In this paper we show that under the assumption of the generalized Riemann hypothesis the following decision problems are in NP ⊃ co-NP: 1.1. Is a given ideal A of O principal?2.2. Given ideals A1, … , Aκ of O, do their equivalence classes generate the class group of O?3.3. Given ideals A1, … , Aκ of O, is the class group of O the direct product of the cyclic subgroups generated by the equivalence classes of the A1

Numerical algorithms for the computation of the Smith normal form of integral matrices,

Numerical algorithms for the computation of the Smith normal form of integral matrices are described. More specifically, the compound matrix method, methods based on elementary row or column operations and methods using modular or p-adic arithmetic are presented. A variety of examples and numerical results are given illustrating the execution of the algorithms