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, χ a nontrivial Dirichlet character modulo p, and

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

where ζ 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 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.

References

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