Dynamic Galois Theory

Abstract

Given a separable polynomial over a field, every maximal idempotent of its splitting algebra defines a representation of its splitting field. Nevertheless such an idempotent is not computable when dealing with a computable field if this field has no factorization algorithm for separable polynomials. Moreover, even when such an algorithm does exist, it is often too heavy. So we suggest to address the problem with the philosophy of lazy evaluation: make only computations needed for precise results, without trying to obtain a priori complete information about the situation. In our setting, even if the splitting field is not computable as a static object, it is always computable as a dynamic one. The Galois group has a very important role in order to understand the unavoidable ambiguity of the splitting field, and this is even more important when dealing with the splitting field as a dynamic object. So it is not astonishing that successive approximations to the Galois group (which is again a dynamic object) are a good tool for improving our computations. Our work can be seen as a Galois version of the Computer Algebra software D5 (Della Dora et al., 1985).