A classical approach to relative quadratic extensions

Abstract

We show that we can develop from scratch and using only classical language a theory of relative quadratic extensions of a given number field K which is as explicit and easy as for the well-known case that K is the field of rational numbers. As an application we prove a reciprocity law which expresses the number of solutions of a given quadratic equation modulo an integral ideal $a$ of K in terms of $a$ modulo the discriminant of the equation. We study various L-functions associated to relative quadratic extensions. In particular, we define, for totally negative algebraic integers Δ of a totally real number field K which are squares modulo 4, numbers $H(\Delta ,K)$, which share important properties of classical Hurwitz class numbers. In an appendix we give a quick elementary proof of certain deeper properties of the Hilbert symbol on higher unit groups of dyadic local number fields.