Reciprocity via Reciprocants

TL;DR

This paper explores the concept of reciprocants, a canonical square root of the resultant of reciprocal polynomials, and demonstrates its application in providing an elegant proof of the Law of Quadratic Reciprocity.

Contribution

It introduces the reciprocant as a new mathematical construct analogous to the Pfaffian for skew-symmetric matrices, linking it to classical number theory results.

Findings

Reciprocants serve as canonical square roots of resultants for reciprocal polynomials.

Computing reciprocants of cyclotomic polynomials offers a concise proof of quadratic reciprocity.

The work establishes a novel connection between algebraic constructs and number theory.

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.