Let $$K=\mathbb {Q}(\sqrt{D})$$ K = Q ( D ) be a real quadratic field. We consider the additive semigroup $$\mathcal {O}_K^+(+)$$ 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 $$\sqrt{D}$$ D . We also characterize all uniquely decomposable integers in K and estimate their norms. Using these results, we prove that the semigroup $$\mathcal {O}_K^+(+)$$ O K + ( + ) completely determines the real quadratic field K .

中文翻译：

---

完全正二次整数的加法结构

设 $$K=\mathbb {Q}(\sqrt{D})$$ K = Q ( D ) 是一个实二次域。我们考虑 K 中全正整数的加性半群 $$\mathcal {O}_K^+(+)$$ OK + ( + ) 并确定其生成元（不可分解整数）和关系；它们可以用 $$\sqrt{D}$$ D 的周期性连分数很好地描述。我们还描述了 K 中所有唯一可分解的整数并估计它们的范数。使用这些结果，我们证明半群 $$\mathcal {O}_K^+(+)$$ OK + ( + ) 完全确定了实二次场 K 。