In mathematics, the Chowla-Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and published independently by Sarvadaman Chowla and Louis Mordell, around 1951.

In detail, if p is a prime number, χ a Dirichlet character modulo p, and

G(χ) = Σ χ(a)ζ^a

where ζ is a primitive p-th root of unity in the complex numbers, then

G(χ)/|G(χ)|

is a root of unity if and only if χ is the quadratic residue symbol modulo p. The 'if' part was known to Gauss: the contribution of Chowla and Mordell was the 'only if' direction. The ratio in the theorem occurs in the functional equation of L-functions.

Reference

Gauss and Jacobi Sums by Bruce C. Berndt, Ronald J. Evans and Kenneth S. Williams, Wiley-Interscience, p.53.