Semester: Fall Instructor: Vadim Vologodsky Course description: The beginnings of number theory can be traced to Diophantine equations: polynomial equations such that only the integer solutions are sought or studied. Surprisingly this is a highly structured part of mathematics: there are general results and conjectures which have many concrete nontrivial corollaries. Number theory uses tools from algebra, analysis, and topology. The course covers some of the most important results obtained by the beginning of the 20th century. Prerequisites: basic abstract algebra (linear algebra, rings, groups, the Galois the- ory) and elementary complex analysis. Curriculum: Finite fields Integers represented by binary quadratic forms Quadratic reciprocity law Division rings over number fields The ideal class group Dirichlet's theorem on units in number fields Local fields Hasse-Minkowski theorem Dirichlet's theorem on primes in arithmetic progressions Analytic class number formula Textbooks: J.P. Serre, A course in Arithmetic. Springer (1973) Z.I. Borevich and I. R. Shafarevich, Number Theory. Academic Press Inc(1966) Algebraic Number Theory, Proceedings of an Instructional Conference Orga- nized by the London Mathematical Society, Edited by Cassels and Frohlich (1967)