Chowla–Mordell theorem

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, $$\chi$$ a nontrivial Dirichlet character modulo $$p$$, and


 * $$G(\chi)=\sum \chi(a) \zeta^a$$

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


 * $$\frac{G(\chi)}{|G(\chi)|}$$

is a root of unity if and only if $$\chi$$ 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.