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Algebraic theories with definable Skolem functions. (English) Zbl 0596.03032
A well-known example of a theory with built-in Skolem functions is first- order Peano arithmetic. The theory of real closed fields has a definitional extension with built-in Skolem functions. Let us say that a theory has definable Skolem functions if it has a definitional extension with built-in Skolem functions. Let T admit quantifier elimination. Then T has definable Skolem functions if and only if each model A of \(T_{\forall}\) has an extension \(A_ 1\) which is algebraic over A and rigid over A. The theory of p-adically closed fields has definable Skolem functions.

MSC:
03C60 Model-theoretic algebra
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[1] Une thĂ©orie de Galois imaginaire 48 pp 1151– (1983)
[2] On definable subsets of p-adic fields 41 pp 605– (1976)
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[4] DOI: 10.1007/BF01218373 · Zbl 0445.16020 · doi:10.1007/BF01218373
[5] Model theory (1973)
[6] DOI: 10.1090/pspum/012/0257030 · doi:10.1090/pspum/012/0257030
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