A classical approach to relative quadratic extensions

BOYLAN H., Skoruppa N.

Journal of Algebra, vol.669, pp.243-272, 2025 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 669
  • Publication Date: 2025
  • Doi Number: 10.1016/j.jalgebra.2025.02.001
  • Journal Name: Journal of Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.243-272
  • Keywords: Algebraic number theory, Discriminants of relative quadratic extensions, Quadratic congruences in number fields, Relative quadratic extensions
  • Istanbul University Affiliated: Yes

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(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.