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Idéaux et types sur les corps séparablement clos. (Ideals and types over separably closed fields). (French) Zbl 0678.03016
It is known that the theory of separably closed fields of fixed characteristic and degree of imperfection is complete, stable and, in case of non-perfect fields, not superstable. The types are described here in terms of ideals of polynomial rings in infinitely many variables. This allows to determine the generic type, to describe forking and to give natural notions of rank, to show that d.o.p. holds, and not f.c.p., and that, in case of a finite degree of imperfection, the expansion of the language via finitely many constants admits elimination of imaginaries. This paper aims at being self-contained: the non-elementary stability definitions and results are recalled in the course of the text. This article together with “The dimensional order property for separably closed fields” by Z. Chatzidakis, G. Cherlin, S. Shelah, G. Srour and C. Wood [Lect. Notes Math. 1292, 78-88 (1987; Zbl 0645.03029)] leaves few remaining open questions. Among those the problem of a type of rank \(\omega^ 2\).
Reviewer: F.Delon

MSC:
03C60 Model-theoretic algebra
12F10 Separable extensions, Galois theory
12L12 Model theory of fields
03C45 Classification theory, stability and related concepts in model theory
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