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 shall 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 size of the coefficients. Finally, we discuss some applications of p-adic numbers, such as the rationality of the zeta function for hypersurfaces over finite fields, as well as solving Diophantine equations and determining zeroes of recurrence sequences.

**Prerequisites**

Basics of groups, rings, and fields (see e.g. Chapters II, III, IV, VII, and VIII of Lang's "undergraduate algebra", which contains much more than needed).

Very basic notions of topology (see e.g. Chapters 2 and 3 of Munkres' "Topology" (second edition), which again contains much more than needed).

- Docent: Frits Beukers
- Docent: Sander Dahmen