Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy.
Jacobi sums J lie in smaller cyclotomic fields than do the nontrivial Gauss sums g. The summands of J(χ, ψ) for example involve no pth root of unity, but rather involve just values which lie in the cyclotomic field of (p − 1)th roots of unity.
When χ is the Legendre symbol, In general the values of Jacobi sums occur in relation with the local zeta-functions of diagonal forms.
The result on the Legendre symbol amounts to the formula p + 1 for the number of points on a conic section that is a projective line over the field of p elements.
Indeed, through the Hasse–Davenport relation of the late 20th century, the formal properties of powers of Gauss sums had become current once more.