List of Solar System objects by size: Difference between revisions

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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.
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In detail, if <math>p</math> is a prime number, <math>\chi</math> a nontrivial [[Dirichlet character]] [[Modulo operation|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 number]]s, 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 [[Carl Friedrich Gauss|Gauss]]: the contribution of Chowla and Mordell was the 'only if' direction. The ratio in the theorem occurs in the [[functional equation (L-function)|functional equation of L-functions]].
 
==References==
* ''Gauss and Jacobi Sums'' by [[Bruce C. Berndt]], Ronald J. Evans and Kenneth S. Williams, Wiley-Interscience, p.&nbsp;53.
 
{{DEFAULTSORT:Chowla-Mordell theorem}}
[[Category:Cyclotomic fields]]
[[Category:Zeta and L-functions]]
[[Category:Theorems in number theory]]
 
[[fi:Chowlan–Mordellin lause]]

Latest revision as of 01:17, 8 January 2015

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