Redei reciprocity, governing fields, and negative Pell

TL;DR

This paper explores the Rdei reciprocity symbol's properties, its role in understanding 8-ranks of class groups, and its implications for the negative Pell equation's solvability, advancing number theory insights.

Contribution

It provides an improved definition of Rdei's trilinear symbol, elucidates its reciprocity, and connects it to Frobenius conditions and negative Pell equation conjectures.

Findings

The Rdei symbol's reciprocity governs 8-ranks of class groups.

Frobenius conditions determine 8-ranks based on prime divisors.

Progress towards negative Pell equation density conjecture is supported.

Abstract

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by R\'edei in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are governed by Frobenius conditions on the primes dividing the discriminant, a fact used the recent work of A. Smith. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation.