Indecomposable Algebraic Integers

TL;DR

This paper investigates the properties of the ring of algebraic integers viewed as a lattice, focusing on indecomposable elements and establishing the decidability of determining whether an algebraic integer is indecomposable.

Contribution

It proves several properties of the ring of algebraic integers as a lattice and establishes the decidability of indecomposability for algebraic integers.

Findings

Properties of the ring of algebraic integers as a lattice are established.

Decidability of the indecomposability of algebraic integers is proven.

Connections between indecomposable elements and lattice properties are discussed.

Abstract

In algebraic number theory, the finiteness of the Picard group of an order in a number field is generally proved via a lattice argument: the order forms a lattice and every ideal class contains an integral ideal with a small enough non-zero element. In this work we will consider $\overline{\mathbb{Z}}$, the ring of algebraic integers, which is a lattice in a similar sense, and we will treat this lattice as intrinsically interesting. We will prove several properties of $\overline{\mathbb{Z}}$ and state some open problems. Many of these properties have connections to the indecomposable elements of the lattice, and our main result regards the decidability of the question whether an algebraic integer is indecomposable.