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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An elementary proof of the law of quadratic reciprocity over function fields


Authors: Chun-Gang Ji and Yan Xue
Journal: Proc. Amer. Math. Soc. 136 (2008), 3035-3039
MSC (2000): Primary 11R58; Secondary 11A15
Published electronically: April 30, 2008
MathSciNet review: 2407064
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ P$ and $ Q$ be relatively prime monic irreducible polynomials in $ \mathbb{F}_{q}[T]$($ 2\nmid q$). In this paper, we give an elementary proof for the following law of quadratic reciprocity in $ \mathbb{F}_{q}[T]$:

$\displaystyle \left (\frac{Q}{P}\right )\left (\frac{P}{Q}\right )=(-1)^{\frac{\vert P\vert-1}{2}\frac{\vert Q\vert -1}{2} },$

where $ \left (\frac{Q}{P}\right )$ is the Legendre symbol.


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Additional Information

Chun-Gang Ji
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
Email: cgji@njnu.edu.cn

Yan Xue
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
Email: xueyan1981521@163.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09327-1
PII: S 0002-9939(08)09327-1
Keywords: Rational function fields, Legendre symbol, quadratic reciprocity law
Received by editor(s): July 6, 2007
Published electronically: April 30, 2008
Additional Notes: The first author is partially supported by grants No. 10771103 and 10201013 from NNSF of China and Jiangsu planned projects for postdoctoral research funds
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.