Algebraic number fields are finite extensions of the field of the rational numbers. They contain a ring of integers, which is a Dedekind domain, a slightly more general type of ring than a principal ideal domain. Non-zero ideals of the ring of integers up to isomorphism form a finite abelian group, the ideal class group of the number field. The ring of integers is a principal ideal domain iff this group is trivial. A theorem of Dirichlet describes the structure of the group of units of a ring of integers. The behavior of these concepts under field extensions will be treated, especially for Galois extensions. Special attention will be given to abelian number fields.

- Docent: Frans Keune