Redei reciprocity, governing fields, and negative Pell

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.