On the divisibility of the class numbers of quaternion orders

TL;DR

This paper proves that the class number of certain quaternion orders over a number field is always divisible by the class number of the base field, revealing a fundamental divisibility property in algebraic number theory.

Contribution

It establishes a general divisibility result for the class numbers of residually unramified quaternion orders over number fields, extending known cases.

Findings

Class number of residually unramified quaternion orders divisible by base field's class number

Includes Eichler orders as a special case

Provides a new divisibility criterion in algebraic number theory

Abstract

Let $F$ be a number field, and $D$ be a quaternion $F$-algebra. We show that the class number of any residually unramified $O_F$-order (e.g. an Eichler order) in $D$ is divisible by the class number of $F$.