AI 中文总结
该研究针对特征0且超越次数≥2的代数闭域上的不可约曲线，证明了存在代数无关坐标的K-点及对应曲线的此类K-点，进而构造了对应非阿基米德实闭域。
AI 中文摘要
设K为特征0且超越次数至少为2的代数闭域，C⊂K²是定义在K上但不定义在Q代数闭包上的不可约曲线。存在C的K-点(x,y)，使得x与y代数无关。此外，若C₀和C₁是两条此类曲线，且它们之间存在定义在K上的有限对一代数对应，则存在对应的K-点(x₀,y₀)∈C₀和(x₁,y₁)∈C₁，满足x₀与y₀代数无关、x₁与y₁代数无关。我们利用后一结果构造了对所有2≤κ≤ℵ₁，超越次数为κ且无非平凡自同构的非阿基米德实闭域。
英文摘要
Let $K$ be an algebraically closed field of characteristic 0 and transcendence degree at least 2. Let $C\subset K^2$ be an irreducible curve defined over $K$ but not defined over the algebraic closure of $\mathbb Q$. There is $(x ,y)$ a $K$-point of $C$ such that $x$ and $y$ are algebraically independent. Moreover, if $C_0$ and $C_1$ are two such curves and there is a finite-to-finite algebraic correspondence between them defined over $K$, then there are corresponding $K$-points $(x_0,y_0)\in C_0$ and $(x_1,y_1)\in C_1$ such that $x_0$ and $y_0$ are algebraically independent and $x_1$ and $y_1$ are algebraically independent. We use the latter result to construct non-Archimedean real closed fields of transcendence degree $κ$ with no non-trivial automorphisms for all $2\leκ\le \aleph_1$.
