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

$\displaystyle \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$).


William Stein 2004-05-06