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. E. Becker has generalized the notion of a real closure (by “real closures at
orderings of higher level”), 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).
