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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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.
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.
Real closed to within three points of La Liga leaders Barca thanks to on Saturday's 2 - 0 win at home to Malaga, when Bale came off the bench in the second half and won a penalty to mark his return from injury.
The partnership with NextLife will enable 300 Recycling to offer our clients a real closed loop process, turning their waste into a resource for new products with NextLife's sustainable resin solutions."
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.
While second-placed Real closed the gap, with an income of pounds 156.
As John Monks of the TUC said yesterday, they represent "the last real closed shop".
Firstly it defines a real closed season in November and March, therefore compressing training periods and easing the burden on over-taxed inter- county players.
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.