Dirichlet's Unit Theorem

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 $K$ has $r$ real and $s$ pairs of complex conjugate embeddings, then

$$\O_K^*\approx \mathbf{Z}^{r+s-1} \times T,$$

where $T$ 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 implies that the solutions to Pell's equation $x^2-dy^2=1$ form a free abelian group of rank $1$.