---
tag: 2767J23XHN
kind: concept
name: Jacobi sum
msc: [11T24, 11L05]
specializes: [VTYJ9419C7]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobi_sum
same_as:
  wikidata: Q6119648
  wikipedia: Jacobi sum
---

## Summary

A Jacobi sum is a sum $\sum_{a} \chi(a)\lambda(1 - a)$ over a [[finite field|0ZXZ82FTX2]], 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|EQ6A3ZN48K]].
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|EMVECB93EQ]] and [[quartic reciprocity|W0M20JWC9K]].

## Explanations

### L1 human

Gauss sums involve complex roots of unity of order $p$, which can be awkward.
Jacobi sums use only the values of multiplicative characters, so they live in a smaller ring of roots of unity.
For the Legendre symbol modulo $5$, the Jacobi sum is $\sum_a \left(\frac{a}{5}\right)\left(\frac{1-a}{5}\right) = -1$.
Writing a prime as a sum of two squares, or as $x^2 + 3y^2$, can be done by computing a suitable Jacobi sum.