Computing $p$-Class Group Structure in Real Quadratic Fields: A New Approach

TL;DR

This paper introduces a novel method for computing the class group structure of real quadratic fields by establishing a new relation that connects class numbers, unit group indices, and class field theory, generalizing previous theories.

Contribution

It presents a new relation linking class number, unit group index, and class field theory, extending existing theories to arbitrary prime powers in real quadratic fields.

Findings

Established a new theoretical relation between class number and unit group index.

Generalized Hilbert class field theory for real quadratic fields.

Bridged class field theory with composition laws of binary forms.

Abstract

This article is the first in a series devoted to computing the class groups of real quadratic fields. We present a new relation between the class number and the index of unit groups. This relation generalizes Hilbert class field theory for real quadratic fields and establishes a bridge between class field theory, composition laws of binary forms of degree $p^n$, and ideal classes of order $p^n$, where p is prime and n is an arbitrary positive integer.