Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9493258 | Journal of Algebra | 2005 | 18 Pages |
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kiran S. Kedlaya, Bjorn Poonen,