域的自同构群的轨道
Orbits of automorphism groups of fields
摘要：
我们解决以下一般问题的几个具体方面：一个域K可以有这么多的自同构，自同构群的作用在K的元素有相对较少的轨道？证明了在其自同构群下只有2个轨道的域是有限的。我们扩展的技术证明，以接近一个更广泛的猜想，它问是否自同构群的一个字段在一个子域只能有1000轨道上的补充子域。最后，我们采用类似的方法来分析领域的马尔切夫-诺依曼“广义幂级数“在一个基本领域，这些形式的近反例，我们的猜想时，基本领域的特征为零，但往往落在令人惊讶的远远不足的积极特征。
英文摘要：
We address several specific aspects of the following general question: can a field K have so many automorphisms that the action of the automorphism group on the elements of K has relatively few orbits? We prove that any field which has only finitely many orbits under its automorphism group is finite. We extend the techniques of that proof to approach a broader conjecture, which asks whether the automorphism group of one field over a subfield can have only finitely many orbits on the complement of the subfield. Finally, we apply similar methods to analyze the field of Mal'cev-Neumann "generalized power series" over a base field; these form near-counterexamples to our conjecture when the base field has characteristic zero, but often fall surprisingly far short in positive characteristic.
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