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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Acknowledgements
We are grateful to the anonymous referee for pointing out that we were using the incorrect notion of semigroup presentation and for several other very useful comments that helped us improve the article.
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We gratefully acknowledge support by Czech Science Foundation (GAČR) Grant 17-04703Y (TH,VK) and partial support by Charles University Research Centre Program UNCE/SCI/022 (VK).
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Hejda, T., Kala, V. Additive structure of totally positive quadratic integers. manuscripta math. 163, 263–278 (2020). https://doi.org/10.1007/s00229-019-01143-8
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DOI: https://doi.org/10.1007/s00229-019-01143-8