Real places and ordered fields
Ron Brown · Rocky Mountain Journal of Mathematics · 1971
Let T:F-»RU {oo } be a place from a field F into the real numbers R. We say an ordering F of F (i.e. the set of nonnegative elements of an order making F an ordered field) is associated with r if and only if T(P) = 0.This definition is closely related to one of Lang's (see the appendix below).In this note we relate the set Ord(r) of orderings of F associated with r to the value group of the valuation canonically associated with r.Precisely, Ord(r) is bijective with the dual of the square factor group of the value group.In particular, Ord(r) is finite if and only if the square factor group is finite, in which case they have the same number of elements.(See [2] for orderings, places and valuations.)The next lemma is essentially an interpretation of some results of Lang [3].Its proof gives the "usual" construction for the real-valued place associated with a given ordering.Notice that an ordering P is associated with r if and only if P contains every element of F which T maps to a positive real number.LEMMA.Each ordering is associated with a unique real-valued place, and each real-valued place has associated with it at least one ordering.PROOF.Let P be an ordering of F. The set of elements of F not infinitely large over the rational numbers forms a valuation ring of F [2, p. 272] ; call the valuation ring A. Let cr 0 be the surjective place canonically associated with A (e.g.[2, p. 298] ).Give <T 0 (A) the ordering <J 0 (A D P).Since this ordering is Archimedean, there is an order embedding o^ of (TQ(A) into R. Then a = cr^o is a realvalued place associated with P. Now suppose a ' is any real-valued place associated with P. Let a G F. If a '(a) < oo, then for some natural number n, 0 < a '(n ± a) < oo ?whence n ± a G P (since P is associated with cr'), so a G A. On the other hand, if cr'(a) = °°, then for each natural number n,