After a short historical introduction and a survey on applications of p-adic numbers, the course will focus on p-adic algebra. The "trinity"

constitutes a beautiful chapter in field theory and the model theory of fields. We will stress the similarities in the structure theory and how it is used to obtain model theoretic results, and in such topics as holomorphy rings of function fields and Nullstellensaetze. In passing, we will give a brief introduction to the basic model theoretic notions and tools which are needed in this area of model theoretic algebra.

Here is a tentative outline of the course:

• A short history of, and motivation for, p-adic numbers
• Absolute values and p-valuations
• The field Q_p of p-adic numbers
• Hensel's Lemma, Newton's Lemma, Newton polygons
• The algebraic closure of Q_p and its completion C_p
• Characterization of p-adically closed fields
• The isomorphism theorem for p-adic closures
• The general embedding theorem
• Notions and tools from model theory
• Model theory of p-adically closed fields
• The Kochen Operator and the Kochen Ring
• The Principal Ideal Theorem
• The holomorphy ring of a function field
• The p-adic Nullstellensatz and integral definite functions

• Witt vectors
• Local-Global Principles
• Topics in p-adic analysis
• p-adic interpolation of the zeta-function
• Topological fields
• The model theoretic work of Ax, Kochen and Ershov

Texts:

1. Alexander Prestel - Peter Roquette: Formally p-adic fields, Springer Lecture Notes in Math. 1050, Berlin (1984)
2. Neal Koblitz: p-adic numbers, p-adic analysis, and Zeta-Functions, Springer, New York (1977)
3. Fernando Gouvea: p-adic numbers - An Introduction, Springer, Berlin (1997)
4. Alain M. Robert: A Course in p-adic analysis, Springer, New York (2000)
5. George Bachman: Introduction to p-adic numbers and valuation theory, Academic Press, New York (1964)
6. Kurt Mahler: p-adic numbers and their functions, Cambridge University Press, Cambridge (1981)
7. Paulo Ribenboim: The Theory of Classical Valuations, Springer, New York (1999)
8. Peter Roquette: History of Valuation Theory, Part I, to appear in: Valuation Theory and its Applications, Proceedings of the Valuation Theory Conference Saskatoon 1999, eds. F.-V. Kuhlmann, S. Kuhlmann and M. Marshall, The Fields Institute Communications Series, Amer. Math. Soc.

The students are expected to hand in assignments (at least twice per month) and to participate actively in class.

The final mark will be calculated as follows:
Oral participation: 10 %
Assignments: 40 %
Final Exam: 50 %.

Last update: January 7, 2002