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AN AMAZING IDENTITY OF GAUSS AND JENKINS’ LEMMA

Part of: Elementary number theory Exponential sums and character sums

Published online by Cambridge University Press:  08 November 2022

HENG HUAT CHAN Open the ORCID record for HENG HUAT CHAN [Opens in a new window]  and
SONG HENG CHAN Open the ORCID record for SONG HENG CHAN [Opens in a new window]
HENG HUAT CHAN
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076, Republic of Singapore e-mail: matchh@nus.edu.sg
SONG HENG CHAN*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang link, Singapore 637371, Republic of Singapore
*
e-mail: chansh@ntu.edu.sg
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Abstract

We prove several finite product-sum identities involving the q-binomial coefficient, one of which is used to prove an amazing identity of Gauss. We then use this identity to evaluate certain quadratic Gauss sums and, together with known properties of quadratic Gauss sums, we prove the quadratic reciprocity law for the Jacobi symbol. We end our article with a new proof of Jenkins’ lemma, a lemma analogous to Gauss’ lemma. This article aims to show that Gauss’ amazing identity and the properties of quadratic Gauss sums are sufficient to establish the quadratic reciprocity law for the Jacobi symbol.

Keywords

Gauss sumJacobi symbolquadratic reciprocity lawcyclotomic unit

MSC classification

Primary: 11L05: Gauss and Kloosterman sums; generalizations
Secondary: 11A07: Congruences; primitive roots; residue systems 11A15: Power residues, reciprocity
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 1 , August 2023 , pp. 86 - 98
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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