Implementation of prime decomposition of polynomial ideals over small finite fields

Abstract

An algorithm for the prime decomposition of polynomial ideals over small finite fields is proposed and implemented. Here "small" means that the order of a finite field is small enough to fit in a single machine word. In particular, we are interested in cases where the order is so small that we cannot apply usual methods which is suc- cessful in cases of characteristic 0. To overcome this difficulty, [8] introduced the notion of "separable ideals" and "separable closure of ideals", and we propose a precise algorithm for the prime decomposition of polynomial ideals over small finite fields based on [8]. Our final goal is to develop a practical algorithm for the primary decomposition of a polynomial ideal over a finite field. For this purpose we can apply the "localization technique" of [7], which enables us to extract primary components from prime divisors. This does not depend on the characteristic of the coefficient field. Therefore primary decomposition computations can be efficiently reduced to prime decomposition computations.