A converse to a theorem of Gauss on Gauss sums

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.