Let $k({\bf x})=k(x_1,\ldots ,x_n)$ be the rational function field, and $k\subsetneqq L\subsetneqq k({\bf x})$ an intermediate field. Then, Hilbert's fourteenth problem asks whether the $k$-algebra $A:=L\cap k[x_1,\ldots ,x_n]$ is finitely generated. Various counterexamples to this problem were already given, but the case $[k({\bf x}):L]=2$ was open when $n=3$. In this paper, we study the problem in terms of the field-theoretic properties of $L$. We say that $L$ is minimal if the transcendence degree $r$ of $L$ over $k$ is equal to that of $A$. We show that, if $r\ge 2$ and $L$ is minimal, then there exists $\sigma \in {\mathop{\rm Aut}\nolimits}_kk(x_1,\ldots ,x_{n+1})$ for which $\sigma (L(x_{n+1}))$ is minimal and a counterexample to the problem. Our result implies the existence of interesting new counterexamples including one with $n=3$ and $[k({\bf x}):L]=2$.

中文翻译：

---

希尔伯特的第十四问题和场修改

设 $k({\bf x})=k(x_1,\ldots ,x_n)$ 为有理函数域，$k\subsetneqq L\subsetneqq k({\bf x})$ 为中间域。然后，希尔伯特的第十四个问题询问 $k$-代数 $A:=L\cap k[x_1,\ldots ,x_n]$ 是否有限生成。已经给出了这个问题的各种反例，但是当 $n=3$ 时 $[k({\bf x}):L]=2$ 的情况是开放的。在本文中，我们根据 $L$ 的场论性质研究该问题。如果 $L$ 对 $k$ 的超越度 $r$ 等于 $A$ 的超越度 $r$，我们说 $L$ 是最小的。我们证明，如果 $r\ge 2$ 和 $L$ 是最小的，那么存在 $\sigma \in {\mathop{\rm Aut}\nolimits}_kk(x_1,\ldots ,x_{n+1} )$ 其中 $\sigma (L(x_{n+1}))$ 是最小的，并且是该问题的反例。