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##
Dynamic Galois Theory

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with
Díaz-Toca G, Journal of Symbolic Computation
** 45 ** (2010), 1316-1329.
** 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 factorisation
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 a complete information about the situtation.
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.