Genus theory and the factorization of class equations over $\mathbb{F}_p$

TL;DR

This paper presents a genus theory-based proof of a theorem characterizing primes for which the class equation of an imaginary quadratic field has a linear factor modulo p, leading to a prime decomposition law in Hilbert class fields.

Contribution

It provides a new genus theory-based proof of Stankewicz's theorem, connecting class equations and prime decomposition in Hilbert class fields.

Findings

Characterizes primes with linear factors in class equations

Establishes a prime decomposition law in Hilbert class fields

Offers a genus theory-based proof approach

Abstract

A new proof, depending only on genus theory, is given of a theorem of Stankewicz, which characterizes the primes $p$ for which the class equation $H_D(X)$ of the maximal order of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{D})$ has a linear factor (mod $p$). This yields a prime decomposition law for the primes $p$ with $p \nmid D$ in the real subfield of the Hilbert class field of $K$.