Adeles

The talk is devoted to adeles of fields of algebraic numbers. We consider their applications to the following questions: functional equation of zeta-functions and the study of its poles, proof of finiteness theorems such as the Dirichlet theorem, description of Galois groups. The following notions will be explained: algebraic numbers, $p$-adic numbers, Dedekind zeta-function, ideal class group, Hilbert symbol. Also, we sketch analogies between algebraic numbers, rational functions over finite fields, and meromorphic functions on Riemann surfaces. At the end of the talk higher-dimensional generalizations of adeles will be discussed.