Chowla–Mordell theorem

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, um 1951.

Im Detail, wenn {Anzeigestil p} is a prime number, {Displaystil Chi } a nontrivial Dirichlet character modulo {Anzeigestil p} , und {Anzeigestil G(chi )=sum chi (a)zeta ^{a}} wo {displaystyle zeta } is a primitive {Anzeigestil p} -th root of unity in the complex numbers, dann {Anzeigestil {frac {G(chi )}{|G(chi )|}}} is a root of unity if and only if {Displaystil Chi } is the quadratic residue symbol modulo {Anzeigestil 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. Kategorien: Cyclotomic fieldsZeta and L-functionsTheorems in number theory