In this chapter we will prove Dirichlet's unit theorem,
which is a structure theorem for the group
of units of the ring of integers of a number field. The answer is
remarkably simple: if has real and pairs of
complex conjugate embeddings,
where is a finite cyclic group.
Many questions can be encoded as questions about the structure of the
group of units. For example, Dirichlet's unit theorem explains
the structure the integer solutions to Pell's equation
(see Section 8.2.1).