**Prerequisites**

The course assumes basic linear algebra (vector spaces, subspaces, quotient spaces, dimension, linear maps, matrices), basic analysis in one variable (sequences, series and their convergence properties; continuous/differentiable functions, power series with ratio of convergence), basic topology (notion of topology, closure, connectedness, connected components, completion, metric) and some algebra (groups, rings, fields, quotient groups, quotient rings, Chinese remainder theorem for rings, field extensions, Galois theory for field extensions of finite degree, finite fields). There are many books that cover those; e.g., for the parts in field theory (where the prerequisites are most advanced) one can look Sections 13.1 through 14.4 of David S. Dummit & Richard M. Foote, "Abstract algebra" (3rd edition, John Wiley & Sons, 2003), as well as many other books.

**Aim of the course**

To provide a thorough introduction to p-adic numbers and discuss some of their applications.

We study p-adic fields, which have many applications in number theory and arithmetic algebraic geometry. We start with the construction and basic properties of the field of p-adic numbers, the completion of the rationals for a metric based on a (fixed) prime number p. We then develop the theory of its finite algebraic extensions, its algebraic closure, and the metric completion of the latter. Next, we study power series with coefficients in such fields, which will include some structural statements and conclusions one can draw from looking at the sizes of their coefficients. Finally, we discuss some applications of p-adic numbers, which include solving Diophantine equations, determining zeros of recurrence sequences, and the rationality of the zeta function for hypersurfaces over finite fields.

**Lecturer**prof. Wadim Zudilin

A 2-hour lecture + 1-hour tutorial session are planned weekly

- Docent: Berend Ringeling
- Docent: Wadim Zudilin