Unit group of a number field (reviewed)

The unit group of a number field $K$ is the group of units of the ring of integers of $K$. It is a finitely generated abelian group with cyclic torsion subgroup. A set of generators of a maximal torsion-free subgroup is called a set of fundamental units for $K$.

The unit group of $K$ has as invariants the rank and the regulator of $K$.