Jacobi-Perron algorithm and indecomposable integers in the simplest cubic fields

Abstract

We will focus on indecomposable integers, one particular subset of algebraic integers in totally real extensions of $\mathbb{Q}$. In the case of quadratic fields $\mathbb{Q}(\sqrt{D})$, we can get all of them using the continued fraction of $\sqrt{D}$ or $\frac{\sqrt{D}-1}{2}$.

Following this relation, we will show how to obtain these elements in the simplest cubic fields using the Jacobi-Perron algorithm, which generates one type of multidimensional continued fractions.

Keywords

• indecomposable integers
• Jacobi-Perron algorithm
• the simplest cubic fields