# Chowla–Mordell theorem

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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.

## References

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