Name: Reciprocity via Reciprocants
Creator: Matthew Baker
Keywords: Mathematics - Number Theory, FOS: Mathematics, Number Theory (math.NT)

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 10 Jan 2025Embargo end date: 01 Jan 2023 English Publisher:Informa UK LimitedJournal:The American Mathematical Monthly, volume 132, pages 303-315 (issn: 0002-9890, eissn: 1930-0972,

Authors: Matthew Baker;

doi: 10.1080/00029890.2024.2439819 , 10.48550/arxiv.2309.02512

arXiv: http://arxiv.org/abs/2309.02512

Reciprocity via Reciprocants

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Abstract

The determinant of a skew-symmetric matrix has a canonical square root given by the Pfaffian. Similarly, the resultant of two reciprocal polynomials of even degree has a canonical square root given by their reciprocant. Computing the reciprocant of two cyclotomic polynomials yields a short and elegant proof of the Law of Quadratic Reciprocity.

10 pages. v2: Rewrote Appendix B, fixed miscellaneous typos

Keywords

Mathematics - Number Theory, FOS: Mathematics, Number Theory (math.NT)

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