Real-local fields and the canonical operation of the automorphism group on the space of orderings

Abstract

The object of our investigation is the canonical operation of the automorphism group of a formally real field F on XF , the space of orderings of F. For a naturally distinguished class of formally real fields, the so-called real-local fields, the Baer-Krull-bijection induces on XF the structure of a module over the endomorphism ring of the group of archimedean classes of F. We show that Aut F acts on XF by affinities with respect to that module structure. Subsequently, this “arithmetization” of the operation is exemplarily applied to the question of transitivity (“When can any two orderings of F be transformed into each other by some automorphism of F?"), and to the investigation of the subgroup of Aut F generated by all order automorphism groups of F.