Natural maps are defined here
which allow many questions about E. Becker’s “orderings of higher level” to be
reduced to questions about the value groups of the real-valued places they induce. A
simple construction is given of the set of orderings of higher level which induce a
given real-valued place (this set is bijective with the set of subgroups of the value
group of the place whose factor groups are cyclic of 2-power order). This construction
leads to straightforward valuation-theoretic characterizations of real closed fields and
of real closures of fields at orderings of higher level. The sets of isomorphism classes
of real closures of a field which induce a given real-valued place, a given
ordering of any level, or even a given family of orderings are each explicitly
computed.
