The behavior of chains of orderings under field extensions and places
Ronald Brown · Pacific Journal of Mathematics · 1987
In the Artin-Schreier theory of formally real fields, the isomorphism classes of certain extensions of a formally real field (namely, the real closures) are shown to correspond bijectively with certain arithmetic structures (namely, orderings) which these extensions induce on the base field.£.Becker has generalized the notion of a real closure (by "real closures at orderings of higher level 9 '), and the isomorphism classes of these generalized real closures again correspond bijectively to certain arithmetic invariants they induce on the base field; these are in essence Harman's "chains of orderings".This paper includes a rather complete analysis of the behavior of such chains of orderings up and down both field extensions and places.The analysis of this behavior is reduced to tractable problems in abelian group theory (together with the analysis of the behavior of ordinary orderings under extensions and places).