Rationality of Four-Valued Families of Weil Sums of Binomials

A Weil sum of a binomial is an object that can be written in the form $\sum_{x \in K} \psi(x^s - u x)$, where $K$ is a finite field, $\psi$ is the canonical additive character of $K$, $s$ is a positive integer, and $u$ is an element of $K^{\times}$. Here, we are only concerned with exponents $s$ that are relatively prime to $|K|-1$. The Weil spectrum for the field $K$ and the exponent $s$ is the multiset of all Weil sums as $u$ runs through $K^{\times}$. The values in the Weil spectrum are algebraic integers in the $p$th cyclotomic extension of ${\mathbb Q}$, where $p$ is the characteristic of the field $K$. Interest in these objects stems from many different areas of study, including arithmetic geometry (in which we wish to count the points in algebraic sets over finite fields), cryptography (in which we wish to determine which power permutations of finite fields can resist linear cryptanalysis), digital sequence design (in which we wish to find families of sequences that are not strongly correlated with any shifted versions of all the others), and coding theory (in which we add redundancy to information to avoid miscommunication that can arise from errors in transmission). One aspect of Weil spectra that is studied is the number of distinct values that they contain, with particular interest in those that have few values. Other than trivial examples, spectra have at least three values. It has been shown that if a spectrum has only three distinct values, then those values must be rational integers. In this thesis, it is shown that (with one exception) four-valued spectra consist of rational integers.