Hurkyl
Oct12-05, 04:52 PM
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?
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?