Finite fields made their
first explicit appearance in the grouptheoretic investigations
of the French mathematician Evariste Galois, in 1830. Nowadays
they play an important role in many parts of pure and applied
mathematics.
Concrete computations in a finite field require the availability
of an explicit model for the field. The present lecture series
addresses a number of fundamental issues that arise in the
context of designing such a model. What should, in the first
place, be meant by an "explicit model" for a finite
field? Can such a model be constructed efficiently? And can
it be recognized?
Further issues arise if different models for the "same"
finite field are encountered. Can an identification between
two such models be found efficiently? And if there are more
than two, how can one guarantee the consistency of the several
pairwise identifications found?
Between any two finite fields of the same cardinality there
is an isomorphism, but that isomorphism is not in general
canonically determined. In the algorithmic world the situation
turns out to be better: between any two explicit models for
finite fields of the same cardinality one can efficiently
construct an isomorphism that may for all practical purposes
be called canonical. This surprising result, which may well
have practical implications, was recently proved in collaboration
with Bart de Smit. It depends on the good algorithmic properties
of suitably defined "standard" models for finite
fields.
The lectures address a general mathematical audience, and
they do not presuppose any specialized knowledge. A precise
formulation of the key results requires the language of theoretical
computer science, but the proof techniques are all taken from
algebra and number theory.
