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(adsbygoogle = window.adsbygoogle || []).push({}); Rwith unique orderings.

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

Now, this is confusing to me -- as far as I know,Qis theonlyfield with a unique ordering:

Any ordered field F must containQ(since it's characteristic zero), so we can "build" an ordering on F with transfinite induction by starting withQand 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 elementanywherein the order we want! E.G. if we were to take the transcendental extensionR(x) ofR, we could makexinfinite, or infinitessimally close to any real number we like.

So I don't understand how any field butQcould 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?

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# Hilbert's 17th and uniquely ordered fields

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