Fermat Hypersurfaces Over Finite Fields

Specialeforsvar ved Sarah Diana Christel Larsen

Titel: Fermat Hypersurfaces Over Finite Fields

Abstract: We work over finite fields and their extensions. We determine the cardinality of the solution set of Fermat hypersurfaces. We calculate this cardinality in different ways: using character theory, namely Gauss- and Jacobi sums, and with projective- and algebraic geometry. We consider several special cases of Fermat hypersurfaces before giving a procedure for treating the general case. Based on these results, we prove the law of quadratic reciprocity and the Hasse-davenport relation. Finally  we prove the rationality of Weil's generating function for the cardinality of the solution set of a general Fermat hypersurface

Vejleder:  Dustin Clausen
Censor:    Peter Beelen, DTU