real closed field

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real closed field

[¦rēl ¦klōzd ′fēld]
(mathematics)
A real field which has no algebraic extensions other than itself.
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A real closed field (RCF) K is a field in which -1 is not a sum of squares and every polynomial of odd degree has a root.
In [MR], Mourgues and Ressayre showed that every real closed field has an integer part.
Concerning the theorems mentioned above, the Axiom of Choice turns out to be indeed necessary: In section 4, we construct transitive models of Zermelo-Fraenkel set theory without the Axiom of Choice (ZF) containing a real closed field K, but no integer part of K.
Remark: In order for some of our statements to make sense, we note that the value group and the residue field of a real closed field K, being definable over K, exist in plain ZF.
We now construct a transitive M |= ZF such that, for some K [member of] M, M satisfies that K is a real closed field that has no integer part.
If K is a real closed field, then X [subset or equal to] K is bounded in K iff there is y [member of] K such that y > x for every x [member of] X.
If K is a countable, unbounded, [omega]-homogenous real closed field, then no integer part I of K has a support.
Hence M believes that K' is a real closed field without an integer part.
If K is a real closed field, then x, y - K are called archimedean equivalent, written x ~ y, iff they are archimedean equivalent as elements of the totally ordered abelian group (K, +).
Let K be a real closed field with value group [theta]([K.sup.x]).
It was proved in [Ka] (Theorem 8) that every real closed field has a value group section with respect to the standard valuation.
These proceedings presents 10 papers on such aspects as the stable Galois correspondence for real closed fields, the Morita equivalence between parametized spectra and module spectra, the linearity of fixed-point invariants, homotopy coherent centers versus centers of homotopy categories, recent developments in noncommutative motives, and the category of Waldhausen categories as a closed multicategory.

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