January 30, 2015

Thematic Program on the Foundations of Computational Mathematics
July-December, 2009

Distinguished Lecture Series

September 16-18, 2009
Fields Institute, Room 230
3:30 - 4:30 pm

Hendrik Lenstra, Mathematisch Instituut, Universiteit Leiden

Modelling finite fields

(audio and slides of talks)
Finite fields made their first explicit appearance in the group-theoretic 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.


Speakers in the Distinguished Lecture Series (DLS) have made outstanding contributions to their field of mathematics. The DLS consists of a series of three one-hour lectures.

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