Chowla–Mordell theorem explained
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
G(\chi)=\sum\chi(a)\zetaa
where
is a primitive
-th root of unity in the
complex numbers, then
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.
