Redei reciprocity, governing fields, and negative Pell

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.