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.

References

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