Relations in the 2-Class Group of Quadratic Number Fields

TL;DR

This paper explores the structure of the 2-class group in quadratic number fields, revealing how ideal class relations are linked to cyclic quartic extensions of the rationals.

Contribution

It introduces a new family of ideals representing order 2 classes and connects their relations to specific cyclic quartic extensions, advancing understanding of class group structure.

Findings

Relations between ideal classes are governed by cyclic quartic extensions.

Constructs a family of ideals representing ideal classes of order 2.

Provides a framework linking class group relations to field extensions.

Abstract

We construct a family of ideals representing ideal classes of order 2 in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.