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The model theory of chain-closed fields

Published online by Cambridge University Press:  12 March 2014

M. A. Dickmann
M. A. Dickmann*
Affiliation:
Equipe de Logique Mathématique, Université Paris-VII, 75251 Paris, France
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Extract

The notion of a higher level ordering is a generalization of the usual notion of an order introduced by Becker in the study of sums of even powers in a field; see [1] for a general reference. A precise definition of an ordering of level 2n (level n in the terminology of [1]) is given in Definition 1.1(II) below.

In [1] Becker worked out the extension theory of fields with a higher level ordering and introduced the notion of a (generalized) real closure for such fields. For a survey of (the analog of) the Artin-Schreier theory of fields with a higher level ordering, see [2]. In [10] Jacob proved decidability, completeness and model-completeness (in a suitable language) for the theory of generalized real-closed fields. Jacob's results are in a sense optimal insofar as the nonuniqueness of generalized real closures (see [1, Chapter IV, Theorems 12 and 13]) prevents quantifier elimination results from holding in languages natural from an algebraic point of view.

The following remarkable fact stems from the work of Becker and Harman: a field having a proper ordering (i.e. one which is not just an order) of any level necessarily has an ordering of (exact) level 2n for each integer n ≥ 2, plus two usual orders, and, moreover, a tight connection holds between orderings of two consecutive levels (cf. [9, Corollary 1.4] and [1, Theorem 15, p. 37]).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 921 - 930
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

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