Quadratic fields are the simplest examples of number fields, yet they already exhibit the full richness of the theory of class groups and class numbers. Their study originated with Gauss’s binary quadratic forms, later linked to quadratic fields via Dedekind’s correspondence between narrow ideal classes and proper equivalence classes of forms.

A significant portion of this chapter is devoted to the reduction theory of binary quadratic forms, which, among other applications, enables the algorithmic computation of class numbers via Dedekind’s correspondence.

For definite forms, reduction yields unique representatives for proper equivalence classes. The modular group acts via linear substitution, inducing Möbius transformations on principal roots in the upper half-plane. Each orbit intersects the fundamental domain at a unique point, corresponding to a canonical reduced form.

Indefinite forms are more intricate. Though the modular group similarly acts on principal roots, no natural fundamental domain exists to select unique orbit representatives. Instead, Zagier’s notion of reduction – distinct from Gauss’s – establishes a bijection between reduced indefinite forms and purely periodic negative regular continued fractions. Remarkably, cyclic permutations of the period generate all reduced forms within a proper equivalence class.