关于“域的自同构群的轨道”一文的商榷
Comments On " Orbits of automorphism groups of fields"
摘要：
设$R$是域$k$上的交换$k-$代数.假设$R$是一个诺特无穷整环。$R$的$k-$自同构群，即$Aut_k(R)$以自然的方式作用于$(R-k)$ .本文第一部分研究了当轨道空间$(R-k)/Aut_k(R)$有限时$R$的结构.我们注意到，[1，§2]中的大多数结果（与域无关）在这种情况下也成立.证明了$R$是一个域。第二部分研究了[1]中猜想2.1的一种特殊情况：若$K/k$是一个非平凡域扩张，其中PDMATHAMO是代数闭的，而$\mid (K-k)/Aut_k(K) \mid = 1$则$K$是代数闭的。最后给出了[1，定理1.1]在$K$是在其素子域上生成的情形下的初等证明.
英文摘要：
Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study the structure of $R$ when the orbit space $(R-k)/Aut_k(R)$ is finite.We note that most of the results, not particularly relevent to fields, in [1,§2] hold in this case as well. Moreover, we prove that $R$ is a field. In the second part, we study a special case of the Conjecture 2.1 in [1] : If $K/k$ is a non trivial field extension where $k$ is algebraically closed and $\mid (K-k)/Aut_k(K) \mid = 1$ then $K$ is algebraically closed. In the end, we give an elementary proof of [1,Theorem 1.1] in case $K$ is finitely generated over its prime subfield.
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