Generalizing a Converse to Gauss’s Theorem on Gauss Sums by Zoe Kane ’25

Abstract:

Let f be a function on a finite field of order p such that the image of f consists of nth roots of unity, f(0) = 0, and f(1) = 1. If the Gauss sum of f has magnitude sqrt(p) does it follow that f is a Dirichlet character? If p does not divide n, the answer is yes, yielding a converse to Gauss’s classical result on Gauss sums of Dirichlet characters. If p divides n, then f cannot be a multiplicative character. This thesis examines this set of non-characters with Gauss sum of magnitude sqrt(p). We discuss how these functions can be constructed from Dirichlet characters via an action of the symmetric group, and we conjecture that this construction is exhaustive. If true, this would amount to a generalization of the converse to Gauss’s theorem that removes the non-divisibility criterion, offering a broader characterization of functions with “character-like” Gauss sums.