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 is a prime number, a nontrivial Dirichlet character modulo, and
where is a primitive -th root of unity in the complex numbers, then
is a root of unity if and only if is the quadratic residue symbol modulo. 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.
