A characterization of Jacobi sums

Abstract

Let $\mathbb{M}$ be the group of multiplicative characters of a finite field $\mathbb{F}$, and let $\mathbb{J}(\alpha, \beta)$ be the Jacobi sum, for $\alpha, \beta \in \mathbb{M}$. We observe that the function $\mathbb{J} \colon \mathbb{M} \times \mathbb{M} \to \mathbf{C}$ satisfies three elementary properties. We show that these properties (very nearly) characterize Jacobi sums: if $M$ is an arbitrary non-trivial finite abelian group and $J \colon M \times M \to \mathbf{C}$ is a function satisfying these properties then $M$ is naturally the group of multiplicative characters of a finite field and $J$ is the Jacobi sum.