×

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

Citations:

Zbl 0645.03029
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] C. Berline , Déviation des types dans les corps algébriquement clos , dans Théories stables 3 ( 1980 - 1982 ), Ed. B. Poizat, Université P. et M. Curie - I. H. P., Paris, 1983 . Numdam | Zbl 0512.03016 · Zbl 0512.03016
[2] N. Bourbaki , XI, Algèbre chapitre 5, Corps commutatifs , Hermann, Paris, 1959 .
[3] E. Bouscaren , Dimensional order property and pairs of models , Ann. of pure and applied Logic, to appear. Zbl 0665.03019 · Zbl 0665.03019 · doi:10.1016/0168-0072(89)90001-8
[4] Z. Chatzidakis , G. CHERLIN , S. Shelah , G. SROUR et C. Wood , Orthogonality of types in separably closed fields , dans Classification Theory (Proceedings of Chicago, 1985 ), Springer Verlag, LNM 1292, Berlin, 1987 . Zbl 0645.03029 · Zbl 0645.03029
[5] L. van den Dries , Model Theory of Fields , Thèse, Utrecht, 1978 .
[6] J. Eršov , Fields with a solvable theory , English translation, Sov. Math. Dokl. 8 ( 1967 ), pp. 575-576. Zbl 0153.37201 · Zbl 0153.37201
[7] J. Keisler , Complete theories of algebraically closed fields with distinguished subfields , Michigan Math. J. 11 ( 1964 ), pp. 71-81. Article | MR 31 #3331 | Zbl 0134.00802 · Zbl 0134.00802 · doi:10.1307/mmj/1028999037
[8] S. Lang , Introduction to algebraic Geometry , Interscience Publishers Inc., New York, 1958 . MR 20 #7021 | Zbl 0095.15301 · Zbl 0095.15301
[9] D. Lascar , Ordre de Rudin-Keisler et poids dans les théories stables , Zeitschr. für math. Logik und Grundl. der Math. 28 ( 1982 ), pp. 413-430. MR 84d:03037 | Zbl 0497.03020 · Zbl 0497.03020 · doi:10.1002/malq.19820282704
[10] D. Lascar , Stabilité en théorie des modèles , Monographies de Mathématique 2, Cabay, Louvain-la-Neuve, 1986 . MR 88e:03052 | Zbl 0655.03021 · Zbl 0655.03021
[11] K. Mc Kenna , Some diophantine Nullstellensätze, Model Theory of Algebra and Arithmetic , (Proceedings of Karpacz 1979 ), Springer Verlag, LNM 834, pp. 228-247. MR 83i:12023 | Zbl 0452.13002 · Zbl 0452.13002
[12] B. Poizat , Déviation des types , Doctorat d’état, Paris 6, 1977 .
[13] B. Poizat , Groupes stables avec types génériques réguliers , J. S. L. 48 ( 1983 ), pp. 339-355. MR 85e:03082 | Zbl 0525.03024 · Zbl 0525.03024 · doi:10.2307/2273551
[14] B. Poizat , Une théorie de Galois imaginaire , J. S. L. 48 ( 1983 ), pp. 1161-1170. MR 85e:03083 | Zbl 0537.03023 · Zbl 0537.03023 · doi:10.2307/2273680
[15] B. Poizat , Paires de structures stables , J. S. L. 48, ( 1983 ), pp. 239-249. MR 84h:03082 | Zbl 0525.03023 · Zbl 0525.03023 · doi:10.2307/2273543
[16] B. Poizat , Cours de théorie des modèles , Nur al-Mantiq wal-Ma’rifah, Villeurbanne, 1985 . MR 87f:03084 | Zbl 0583.03001 · Zbl 0583.03001
[17] A. Robinson , Solution of a problem of Tarski , Fund. Math., XLVII ( 1959 ), pp. 179-204. Article | MR 22 #3690 | Zbl 0093.01305 · Zbl 0093.01305
[18] S. Shelah , Stability, the f. c. p., and superstability ... , Ann. Math. Logic 3 ( 1971 ), pp. 271-362. Zbl 0281.02052 · Zbl 0281.02052 · doi:10.1016/0003-4843(71)90015-5
[19] S. Shelah , The lazy model-theoretician’s guide to stability , dans Six Days of Model Theory, Ed. P. Henrard, Castella, Albeuve, 1977 . Zbl 0414.03022 · Zbl 0414.03022
[20] S. Shelah , Classification Theory and the Number of non-isomorphic Models , North Holland, Amsterdam, 1978 . Zbl 0388.03009 · Zbl 0388.03009
[21] S. Shelah , The Spectrum Problem I : \aleph \epsilon -saturated models , the Main Gap, Israel J. Math. 43 ( 1982 ), pp. 324-336. Zbl 0532.03013 · Zbl 0532.03013 · doi:10.1007/BF02761237
[22] G. Srour , The independance relation in separably closed fields , J. S. L. 51 ( 1986 ), pp. 715-725. MR 87m:03047 | Zbl 0633.03029 · Zbl 0633.03029 · doi:10.2307/2274025
[23] G. Srour , Types of rank 1 in the theory of separably closed fields preprint . · Zbl 0633.03029
[24] C. Wood , Notes on the stability of separably closed fields , J. S. L. 44 ( 1979 ), pp. 412-416. MR 81m:03042 | Zbl 0424.03014 · Zbl 0424.03014 · doi:10.2307/2273133
[25] O. Zariski et P. Samuel , Commutative Algebra I , Van Nostrand Company, Princeton, 1958 . MR 19,833e | Zbl 0081.26501 · Zbl 0081.26501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.