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.
See also Dirichlet L-function – Type of mathematical function
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