Additive structure of totally positive quadratic integers

Tomáš Hejda, Vítězslav Kala

Published 2017-06-28Version 1

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We obtain a presentation of the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$; its generators (indecomposable integers) and relations can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$.