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Orbits of automorphism groups of fields

Kiran S. Kedlaya·Bjorn Poonen
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Abstract

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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Kiran S. Kedlaya, Bjorn Poonen. 2005-06-28. Orbits of automorphism groups of fields. https://arxiv.org/abs/math/0407473

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