Additive structure of totally positive quadratic integers

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 $K$.

15 pages, to appear in manuscripta mathematica

References in corpus (4)

• Number fields without n-ary universal quadratic forms
• Universal quadratic forms and elements of small norm in real quadratic fields
• Norms of indecomposable integers in real quadratic fields
• Idempotence of finitely generated commutative semifields

Cited by in corpus (5)

• Universal quadratic forms, small norms and traces in families of number fields
• Universal quadratic forms and indecomposables in number fields: A survey
• Trace and norm of indecomposable integers in cubic orders
• Partitions into powers of an algebraic number
• Triangle Percolation on the Grid