Model Theory of Valued Fields
Instructor: David Marker
Class Meets: MWF 10:00-10:50 ????
Office: 404 SEO
Office Hours: TBA
phone: (312) 996-3069
course webpage: http://www.math.uic.edu/~marker/math512-f18
The valued fields have long been of interest to model theorists. Model theoretic methods have had important applications to number theory and valued fields provide a rich source of examples of model theoretic phenomena. This course will be an
introduction to the model theory and algebra of valued fields. The two main areas of focus will be:
- Henselian valued fields, the work of Ax--Kochen and Ershov on Artin's conjecture,
the model theory of the p-adics including Macintyre's quantifier elimination and, time permitting, Denef's cell-decomposition and p-adic integrals.
- Algebraically closed valued fields, quantifier elimination and elimination of imaginaries
Depending on time and the interests of the audience I hope to introduce some more advanced applications in the work of Hrushovski and Loeser.
Prerequisites The course will be relatively self contained and I will try to keep the model theoretic and algebraic prerequisites to a minimum.
- The model theoretic background can be found in sections 1--6 of my Orsay lecture notes
Model Theory of Algebra and Algebraic Geometry
- I will assume basics on rings and fields including some elementary commutative algebra and Galois theory, but no previous knowledge of valued fields.
References There is no text for the course but I will be preparing lecture notes. There are a number of very useful references.
Last Revised: 5/13/18
- A. Engler and A. Prestel, Valued Fields, Springer 2005.
This is an excellent concise introduction to the algebra of valued fields including a gentle introduction to the model theoretic connections. It is available for download from the UIC Library.
- Z. Chatzidakis,
Theorie des Models des corps values, 2008.
Lecture notes from a course on valued fields.
- L. van den Dries, Lectures On the model theory of valued fields,
Model Theory in Algebra, Analysis and Arithmetic, H. D. Macpherson and C.
Tuffatori ed., Lecture Notes in Mathematics 2011, pp. 55-158 Springer 2014.
Lecture notes from a course on the model theory of valued fields. This entire volume is available on-line via the UIC Library.
- W. Johnson, On the proof of elimination of imaginaries in algebraically closed valued fields
- E. Hrushovski and F. Loeser, Non-Archimedean Tame Topology and Stably Dominated Types, Annals of Mathematics Studies 192, Princeton 2014.