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.
Integer parts of real closed fields are especially interesting as they are known to coincide with models of a certain natural fragment of Peano Arithmetic, namely Open Induction (see [S]).
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.
Let K be a countable, unbounded, [omega]-homogenous real closed field, let [?
His principal goal is to examine the known results on the equivalence theory and related matters such as the Witt and Witt-Grothendieck groups, over the classical fields: algebraically closed, real closed
, finite, local and global.
This collection of Artin's work includes his books Galois Theory, The Gamma Function and The Theory of Algebraic Numbers, and papers on the axiomatic characterization of fields with George Whaples, real fields ("A Characterization of the Field of Real Algebraic Numbers," "The Algebraic Construction of Real Fields" and "A Characterization of Real Closed
Fields") in their first English translation, and the theory of braids.