Dirichlet's Unit Theorem

In this chapter we will prove the main 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 $s$ complex embeddings, then

$$\O _K^*\approx \mathbf{Z}^{r+s-1} \oplus W,$$

where $W$ is the finite cyclic group of roots of unity in $K$. Examples will follow on Thursday (application: the solutions to Pell's equation $x^2-dy^2=1$, for $d>1$ squarefree, form a free abelian group of rank $1$).