2022/08/06 by Boylan, Hatice, Skoruppa, Nils-Peter
#11R11 #11R29 #11S99 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2208.03515
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 \mathfraka of K in terms of \mathfraka 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(Δ,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.