A converse to a theorem of Gauss on Gauss sums

TL;DR

This paper establishes a converse to Gauss's theorem on Gauss sums, showing that a function with a Fourier transform of magnitude 1 at some point must be a nontrivial character, revealing new insights into character sums.

Contribution

It proves a converse to Gauss's theorem on Gauss sums, linking Fourier transform properties to character functions over finite fields.

Findings

Characterization of nontrivial characters via Fourier transform magnitude

Converse to Gauss's theorem on Gauss sums established

Implications for understanding character sums and Fourier analysis in finite fields

Abstract

In this note we prove (under mild hypotheses) that $f$ is a nontrivial character of $\mathbb{F}_p$ if and only if the Fourier transform of $f$ has magnitude 1 somewhere in $\mathbb{F}_p^\times$. This implies a converse to a theorem of Gauss on the magnitude of the Gauss sum, in addition to other consequences.