The tools ?
finite fields, Gauss sums, quasi-Frobenius rings, characters, Fourier transform, arithmetic of cyclotomic fields and... computers
the document ?
It is in french. The introduction of the memory looks like a synthesis of my works. The results of my publications are described in six chapters : Fourier Transforms Boolean functions Gauss sums sequences Finite Fields codes.
The main objective is to give a pedagogical introduction to characters theory applied to the above domains.
many open questions and problems are included.
the talk of today ?
I will explain the influence of the proof of a theorem by Ax in my work. Some applications of Gauss sums will be presented and perspectives will be given.