Units in number fields with complex embeddings Assume that we have an algebraic number field with integers $o$, and with a complex embedding $\iota$.
What can be said about the image $\iota( o^\times)$ under $\iota$? 
Is it discrete? Is infinite?

As has been noted in comments, the important result here is Dirichlet's Unit Theorem. E.g., (taken from Daniel Marcus's Number Fields):

Dirichlet's Unit Theorem. Let $U$ be the group of units in a number ring $\mathcal{O}_K = \mathbb{A}\cap K$ (where $\mathbb{A}$ represents the ring of all algebraic integers). Let $r$ and $2s$ denote the number of real and non-real embeddings of $K$ in $\mathbb{C}$. Then $U$ is the direct product $W\times V$, where $W$ is a finite cyclic group consisting of the roots of $1$ in $K$, and $V$ is a free abelian group of rank $r+s-1$. 

In particular, there is some set of $r+s-1$ units, $u_1,\ldots,u_{r+s-1}$ of $\mathcal{O}_K$, called a fundamental system of units, such that every element of $V$ is a product of the form
$$u_1^{k_1}\cdots u_{r+s-1}^{k_{r+s-1}},\qquad k_i\in\mathbb{Z},$$
and the exponents are uniquely determined for a given element of $V$. 
