Chowla–Mordell theorem
Open-knowledge reference entry
In mathematics, the Chowla–Mordell theorem says that for a Dirichlet character modulo an odd prime, if the argument of its Gaussian sum is a root of unity, then the character must be quadratic. It 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.
Statement
If
p
{\displaystyle p}
is a prime number,
χ
{\displaystyle \chi }
a nontrivial Dirichlet character modulo
p
{\displaystyle p}
, and
G
(
χ
)
=
∑
χ
(
a
)
ζ
a
{\displaystyle G(\chi )=\sum \chi (a)\zeta ^{a}}
where
ζ
{\displaystyle \zeta }
is a primitive
p
{\displaystyle p}
-th root of unity in the complex numbers, then
G
(
χ
)
|
G
(
χ
)
|
{\displaystyle {\frac {G(\chi )}{|G(\chi )|}}}
is a root of unity if and only if
χ
{\displaystyle \chi }
is the quadratic residue symbol modulo
p
{\displaystyle 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.
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