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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? –  DonAntonio Jul 19 '12 at 15:29
    
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! –  plusepsilon.de Jul 19 '12 at 15:31
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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. –  DonAntonio Jul 19 '12 at 15:35
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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.) –  Qiaochu Yuan Jul 19 '12 at 15:44
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The torsion subgroup is exactly the group of roots of unity in the field. –  paul garrett Jul 19 '12 at 16:23

1 Answer 1

up vote 3 down vote accepted

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 $V$.

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Dirichlet\neq Dedekind=) Thank you for providing a reference. –  plusepsilon.de Jul 20 '12 at 7:08
    
@late_learner: Oops; thank you. –  Arturo Magidin Jul 20 '12 at 15:32
    
@Brandon: Double oops... Need to drink my coffee before editing... –  Arturo Magidin Jul 20 '12 at 15:38

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