Jacobi sum

Summary

A Jacobi sum is a sum $\sum_{a} \chi(a)\lambda(1 - a)$ over a finite field, built from two multiplicative characters $\chi$ and $\lambda$. It counts, in a weighted way, the solutions of $x + y = 1$. When $\chi$, $\lambda$ and $\chi\lambda$ are nontrivial, it is a quotient of Gauss sums. Then its absolute value is the square root of the field size. Jacobi sums are used to count points on curves such as $x^n + y^n = 1$ over finite fields and to prove cubic and quartic reciprocity.