Characters, Gauss Sums, and the DFT

Abstract

One of the major problems often encountered in working with finite fields is the transition between their additive and multiplicative structures. We have seen a concrete example of this type of problem when we studied the arithmetics in finite fields in Chap. 8. Depending on the choice of representation, either addition (in some basis representation) or multiplication (using Zech logarithms) is trivial, while the other operation is not at all easy to perform. The same type of problem arises also in theoretical investigations concerning both the additive and multiplicative structure simultaneously, for instance, when proving the existence of primitive normal bases or that of primitive elements with a prescribed value of the trace (into a specified subfield); these two problems will the topic of the final two chapters. Such questions often require the use of tools from Representation Theory which we did not introduce up to now, that is, characters and character sums and, in particular, Gauss sums. These tools will be presented in the current chapter, where we first consider characters of finite abelian groups in general and then specialize to the case of finite fields and introduce the basic properties of Gauss sums. We then give a few interesting applications of the quadratic character; in particular, we shall prove the law of quadratic reciprocity and consider solutions of quadratic equations in several variables over finite fields with odd characteristic. Following this, we shall prove three more advanced identities for Gauss sums; we will also obtain an interesting connection to the eigenvalues of the matrix of the Discrete Fourier Transform defined in Sect. 7.4. Finally, we extend the DFT to abelian groups in general and present some applications to abelian difference sets and periodic sequences.