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arXiv · 1706.09178

Additive structure of totally positive quadratic integers

Tomáš Hejda·Vítězslav Kala
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Abstract

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they 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 $K$.

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Tomáš Hejda, Vítězslav Kala. 2017-06-28. Additive structure of totally positive quadratic integers. https://doi.org/10.1007/s00229-019-01143-8

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