Harbingers of Artin's Reciprocity Law. I. The Continuing Story of Auxiliary Primes

TL;DR

This paper reviews the historical development of auxiliary primes in reciprocity laws, highlighting how Gauss's genus theory can address gaps in Legendre's proof of quadratic reciprocity.

Contribution

It demonstrates that the gap in Legendre's proof of quadratic reciprocity can be closed using Gauss's genus theory, connecting historical methods with modern insights.

Findings

Historical overview of auxiliary primes in reciprocity laws

Closure of Legendre's proof gap using genus theory

Connection between classical and modern number theory methods

Abstract

In this article we present the history of auxiliary primes used in proofs of reciprocity laws from the quadratic to Artin's reciprocity law. We also show that the gap in Legendre's proof can be closed with a simple application of Gauss's genus theory.