Hilbert's 17th and uniquely ordered fields

[Hurkyl]

Jacobson's Basic Algebra II has a section on Hilbert's 17th problem (p. 660), and gives the Theorem of Artin which involves subfields of R with unique orderings.

In the text, it says "Examples of fields having a unique ordering are Q, R, and any number field that has only one real conjugate field".

Now, this is confusing to me -- as far as I know, Q is the only field with a unique ordering:

Any ordered field F must contain Q (since it's characteristic zero), so we can "build" an ordering on F with transfinite induction by starting with Q and build F by a (transfinite) sequence of algebraic and transcendental extensions.

Any algebraic extension has a nontrivial Galois group, and we have at least one ordering for each element of the Galois group.

The case of a transcendental extension is even worse: we can place the new element anywhere in the order we want! E.G. if we were to take the transcendental extension R(x) of R, we could make x infinite, or infinitessimally close to any real number we like.


So I don't understand how any field but Q could have a unique ordering. I've read through the chapter in Jacobson, but have been able to find anything that would explain my problem. Anyone out there know?

 

[fresh_42]

Now, this is confusing to me -- as far as I know, Q is the only field with a unique ordering:

No, this is not true. E.g. if $K$ is the quotient field of a ring $R$ and $R$ is ordered, then there is a unique order on $K$ which extents the ordering in $R$. This gives quite a few uniquely ordered fields.