**Aim**

The course provides a thorough introduction to algebraic number theory. It treats the basic laws of arithmetic that are valid in subrings of algebraic number fields.

**Description**

Introduction to algebraic numbers and number rings. Ideal factorization, finiteness results on class groups and units, explicit computation of these invariants. Special topics: binary quadratic forms, the number field sieve, valuations and completions, local fields, introduction to class field theory and reciprocity laws, density theorems.

**Organization**

The class is taught every Tuesday morning at the Vrije Universiteit in Amsterdam: two hours (10.15-12.00) of lectures and one hour (12.15-13.00) devoted to exercises.

**Examination**

The final grade is exclusively based on the results obtained for the weekly homework assignments.

**Literature**

There will be course notes and homework exercises. Several books entitled `Algebraic number theory’, such as those by Stewart & Tall, E. Weiss, S. Lang, J. Neukirch or Cassels & Fröhlich, can be profitably consulted. See also the course notes`

Number rings’ at

http://websites.math.leidenuniv.nl/algebra/

**Prerequisites**

Undergraduate algebra, i.e., the basic properties of groups, rings, and fields, including Galois theory. This material is covered in first and second year algebra courses in the bachelor program of most universities. See http://www.math.leidenuniv.nl/algebra for the course notes used in Leiden and Delft.

**Prerequisites**

Algebra 1, 2, 3

**Required for**

Master projects in algebraic number theory or arithmetic geometry

**Classes at**

VU Amsterdam