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- W2003038601 abstract "The complete, model-complete theories of pseudo-algebraically closed fields were characterized completely in [11]. That work constituted the first step towards determining all the model-complete theories of fields in the usual language of fields. In this paper the second step is taken. Namely, the methods of [11] are extended to characterize the complete, model-complete theories of pseudo-real closed fields and pseudo- p -adically closed fields. In order to unify the treatment of these two types of fields, the relevant properties of real closed ordered fields and p -adically closed valued fields are abstracted. The subsequent investigation of model-complete theories of fields is based entirely on these properties. The properties were selected in order to solve three problems: (1) finding universal theories with the joint embedding property, (2) finding first order conditions in the usual language of fields which are necessary and sufficient for a polynomial over a field to have a zero in a formally real or formally p -adic extension of that field, and (3) finding subgroups of Galois groups whose fixed fields are formally real or formally p -adic. This paper is related to, and uses in §1 but not in the other sections, parts of K. McKenna's work [8] on model-complete theories of ordered fields and p -valued fields. However, the results herein are not direct consequences of his work, both because these results apply to a more general situation and because they use a different formal language. Concerning the latter point, in some instances, such as real closed ordered fields and p -adically closed valued fields, model-complete theories in expanded languages do yield model-complete theories of ordinary fields other than theories of pseudo-algebraically closed fields. However, in other cases, such as differentially closed fields, this is not so." @default.
- W2003038601 created "2016-06-24" @default.
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- W2003038601 date "1983-12-01" @default.
- W2003038601 modified "2023-09-24" @default.
- W2003038601 title "Model-complete theories of formally real fields and formally p-adic fields " @default.
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- W2003038601 doi "https://doi.org/10.2307/2273676" @default.
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