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Hilbert's 17th and uniquely ordered fields

  1. Oct 12, 2005 #1

    Hurkyl

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    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. :frown: 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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