### p-Adic Numbers - 8EC - M1

Prerequisites

The course assumes basic linear algebra (vector space, subspace, quotient space, dimension, linear map, matrix), basic analysis in one variable (sequences, series and their convergence properties; continuous or differentiable functions, power series with ratio of convergence), basic topology (notion of topology, closure, connectedness/connected component, completion, metric), and some algebra (definitions of 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 the most advanced) one can look Sections 13.1 through 14.4 of Dummit&Foote, Abstract algebra, third edition), 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 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 sizes of the coefficients. Finally, we discuss some applications of p-adic numbers, which may include the rationality of the zeta function for hypersurfaces over finite fields, solving Diophantine equations, determining zeroes of recurrence sequences.