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)