×

Model theory of fields. (English) Zbl 0911.12005

Lecture Notes in Logic. 5. Berlin: Springer-Verlag. ix, 154 p. (1996).
The merit of these Lecture Notes is to present some classical notions from algebraic, analytic and differential geometry in a model-theoretic setting. Conversely several results from model theory are interpreted into classical language. The book consists of four papers, which survey the model theory of some important elementary classes of fields. Properties such as model completeness, elimination of quantifiers, stability and elimination of imaginaries are reviewed.
In Chapter 1, algebraically closed and real closed fields are considered. Typically illustrating the spirit of these notes, the Nullstellensatz is obtained as a corollary to elimination of quantifiers for algebraically closed fields, and the theorem of Chevalley is presented as a geometric restatement of quantifier elimination. The Zariski spectrum is shown to have its model-theoretic analogue (the space of types) and the relation between the (classical) dimension and the Morley rank of a variety is worked out. The theorem of Tarski-Seidenberg is presented as a geometric restatement of quantifier elimination for real closed fields, and A. Robinson’s model-theoretic solution to Hilbert’s 17th Problem is exposed. The notion of o-minimality and the properties of definable functions in that setting are reviewed, and cell decomposition is presented as a higher dimensional version of Tom’s Lemma. The chapter ends with a review of the geometry of strongly minimal sets, Zil’ber’s conjecture and its refutation by Hrushovski.
Chapter 2 reviews some basic differential algebra, followed by basic model theory of differentially closed fields (DCF) (in characteristic 0). Seidenberg’s Differential Nullstellensatz is obtained as a corollary to elimination of quantifiers. The existence and uniqueness of differential closures is obtained by first showing that DCF is \(\omega\)-stable, and then using a result by Morley and Shelah on the existence and uniqueness of prime models for \(\omega\)-stable theories. The non-minimality of differential closures is discussed, and Rosenlicht’s proof of this fact (using differential forms) is given. In section 7, DCF is shown to have DOP, which implies (by a theorem of Shelah) that there are \(2^{\kappa}\) non-isomorphic DCF of cardinality \(\kappa\), for uncountable cardinal \(\kappa\). The last 3 sections are devoted to differential Galois theory for strongly normal extensions. The author first considers the Galois-correspondence for a Picard-Vessiot extension \(L| K\) and provides a proof that the differential Galois group \(G(L| K)\) in this case is a linear algebraic group over the constant field. Next he considers Kolchin’s strongly normal extensions. He establishes that \(G(L| K)\) is definable in the algebraically closed constant field, and uses a result of van den Dries to deduce Kolchin’s theorem that \(G(L| K)\) is an algebraic group over the constant field (if \(K\) is algebraically closed). In the last section, the author discusses the open question whether a superstable differential field is necessarily differentially closed, and presents a partial result due to Pillay and Sokolovic.
Chapter 3 povides a proof that the number of non-isomorphic countable models of DCF is \(2^{\aleph _0}\), avoiding the Zariski-geometry machinery. The main ingredient of this proof is to find some strongly regular non-isolated type orthogonal to the empty set. This is done using an example of Manin on elliptic curves.
Chapter 4 studies separably closed fields (in characteristic \(p\not=0\)). Their model theory, mainly worked out by Delon, was used in Hrushovski’s proof of the Mordell-Lang conjecture for function fields. Completeness, model-completeness and elimination of quantifiers results (in the case of finite degree of imperfection), all mainly due to Ershov, are discussed. A description of types in terms of ideals, a proof of Wood’s result that the theory is stable but not superstable, as well as a discussion of Forking and DOP in this context are given. The last section presents equivalent formulations of elimination of imaginaries for the theory of a field, and points out the connection to the existence of canonical bases for types.
Unfortunately, the book is full of misprints, especially in the first two chapters, and readers not familiar with the material presented could be seriously misled by them.

MSC:

12L12 Model theory of fields
12-02 Research exposition (monographs, survey articles) pertaining to field theory
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03C60 Model-theoretic algebra
03C10 Quantifier elimination, model completeness, and related topics
12H05 Differential algebra
03C45 Classification theory, stability, and related concepts in model theory
PDFBibTeX XMLCite