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Additive structure of totally positive quadratic integers

  • Tomáš HejdaEmail author
  • Vítězslav Kala
Article

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.

Mathematics Subject Classification

11R11 11A55 20M05 20M14 

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Algebra, FMPCharles UniversityPragueCzechia
  2. 2.Department of Mathematics, FCEUniversity of Chemistry and TechnologyPragueCzechia

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