AI 中文总结
该研究针对数域K，在特定条件下证明其可表为极大序中两单位之和的正整数集合有限且可有效计算，部分解决了开放问题，并以S₅群的最小全实五次域为例验证方法。
AI 中文摘要
设K为次数为D的数域，其极大序为𝒪_K。我们证明，在K满足特定条件下（当D为奇数或D≥3且K为本原型时，该条件恒成立），可表示为𝒪_K^*中两个单位之和的正整数集合N_K是有限且可有效计算的。该结果部分解决了Tinková、Yatsyna及第一作者提出的开放问题。我们以具有S₅伽罗瓦群的最小全实五次域K为例，通过显式计算N_K阐释了所提方法。
英文摘要
Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a finite effectively computable set. This result partially resolves an open problem posed by Tinková, Yatsyna, and the first author. We illustrate our method by explicitly computing $N_K$ for the smallest totally real quintic field $K$ with Galois group $S_5$.
Comments20 pages; comments are welcome!
