Redei reciprocity, governing fields and negative Pell

Abstract We discuss the origin, an improved definition and key reciprocity property of trilinear symbol introduced by Rédei [16] in study 8-ranks narrow class groups quadratic number fields. It can be used to show that such are ‘governed’ Frobenius conditions on primes dividing discriminant, a fact recent work A. Smith [18, 19]. In addition, we explain its impact progress towards proving my conjectural density for solvability negative Pell equation $x^2-dy^2=-1$ .