A precise result on the arithmetic of non-principal orders in algebraic number fields

TL;DR

This paper investigates the arithmetic properties of non-principal orders in algebraic number fields, focusing on half-factoriality, using a novel semigroup theoretical approach to extend known results from principal orders.

Contribution

It introduces a new semigroup theoretical method to analyze half-factoriality and arithmetical properties of non-principal orders, which are less understood than principal orders.

Findings

Characterizes half-factoriality in non-principal orders

Develops a new semigroup approach for non-principal orders

Extends explicit arithmetic results beyond principal orders

Abstract

Let $R$ be an order in an algebraic number field. If $R$ is a principal order, then many explicit results on its arithmetic are available. Among others, $R$ is half-factorial if and only if the class group of $R$ has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.