A characterization of Jacobi sums

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 $\mathbb{J}\text{\hspace{0.17em}⁣}:\mathbb{M}\times \mathbb{M}\to \mathbf{C}$ 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 $J\text{\hspace{0.17em}⁣}:M\times M\to \mathbf{C}$ 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.