Chowla–Mordell theorem

Template:Short description 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 <math>p</math> is a prime number, <math>\chi</math> a nontrivial Dirichlet character modulo <math>p</math>, and

<math>G(\chi)=\sum \chi(a) \zeta^a</math>

where <math>\zeta</math> is a primitive <math>p</math>-th root of unity in the complex numbers, then

<math>\frac{G(\chi)}{|G(\chi)|}</math>

is a root of unity if and only if <math>\chi</math> is the quadratic residue symbol modulo <math>p</math>. 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.