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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?

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    What have you done so far? What do you know about the...ring?, group?, field?...of units in a number field? What's your motivation for this exercise?2012-07-19
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    I do not want to make more restriction to the number field. It is not an exercise I read somewhere. So I have no clue how to approach it. It might be trivial!2012-07-19
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    I asked because the great Dirichlet's Theorem on Units says *the group* of units in a number field is finitely generated of rank $\,r_1+r_2-1\,$ ,with $\,r_1=\,$ number of real embeddings of the field, $\,2r_2=\,$number complex embeddings.2012-07-19
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    So it is discrete as every unit must be torsion, I guess. Thanks that was fast help. Thx2012-07-19
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    I should delete the question then probably soon?!2012-07-19
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    It is not true that every unit must be torsion (take $1 + \sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$). Don Antonio's statement of the unit theorem does not describe the torsion subgroup. (The torsion subgroup is just some finite abelian group. This is not very hard to see.)2012-07-19
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    @late_learner, first read Yuan's comment. If you're interested in the torsion subgroups of the units, which of course is also of finite rank, then you can deduce what you wrote in your above comment. Second, it's not a good idea to delete a question. Better, you write an answer to your own question!2012-07-19
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    The torsion subgroup is exactly the group of roots of unity in the field.2012-07-19

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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 $V$.