Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
Number Theory
2025-03-19 v2
Abstract
We study a new connection between multidimensional continued fractions, such as Jacobi--Perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables of all signatures in Ennola's family of cubic fields, and use them to determine the Pythagoras numbers. Second, we compute a number of periodic JPA expansions, also in Shanks' family of simplest cubic fields. Finally, we compare these expansions with indecomposables to formulate our conclusions.
Keywords
Cite
@article{arxiv.2303.00485,
title = {Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables},
author = {Vítězslav Kala and Ester Sgallová and Magdaléna Tinková},
journal= {arXiv preprint arXiv:2303.00485},
year = {2025}
}