Background: a field is *formally real* if -1 is not a sum of squares of elements in that field. An *ordering* on a field is a linear ordering which is (in exactly the sense that you would guess if you haven't seen this before) compatible with the field operations.


It is immediate to see that a field which can be ordered is formally real. The converse is a famous result of Artin-Schreier. (For a graceful exposition, see Jacobson's Basic Algebra. For a not particularly graceful exposition which is freely available online, see [http://alpha.math.uga.edu/~pete/realspectrum.pdf.](http://alpha.math.uga.edu/%7Epete/realspectrum.pdf.))


The proof is neither long nor difficult, but it appeals to Zorn's Lemma. One suspects that the reliance on the Axiom of Choice is crucial, because a field which is formally real can have many different orderings (loc. cit. gives a brief introduction to the *real spectrum* of a field, the set of all orderings endowed with a certain topology making it a compact, totally disconnected topological space).


Can someone give a reference or an argument that AC is required in the technical sense (i.e., there are models of ZF in which it is false)? Does assuming that formally real fields can be ordered recover some weak version of AC, e.g. that any Boolean algebra has a prime ideal? (Or, what seems less likely to me, is it equivalent to AC?)



How much choice is needed to show that formally real fields can be ordered?
This is equivalent (in ZF) to the Boolean Prime Ideal Theorem (which is equivalent to the Ultrafilter Lemma).


Reference:
R. Berr, F. Delon, J. Schmid, Ordered ﬁelds and the ultraﬁlter theorem, Fund Math 159 (1999), 231-241. [online](http://matwbn.icm.edu.pl/ksiazki/fm/fm159/fm15933.pdf)


