The number of algebraic integer within the unit disk Let $K$ be a number field (I am mostly interested in the case $K=\Bbb Q(\zeta_n)$ is a cyclotomic field).
Let $\alpha$ be an algebraic integer in $K$. I would like to know


*

*whether there are only finitely many such $\alpha$ whose absolute value is less than 1.

*If so, is there any explicit bound?

*How about if we restrict to real algebraic integers with absolute value less than 1?


For 2, if $K=\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is a primitive $n^{th}$ root of unity, then is there a bound in terms of $n$?
 A: Here I'm assuming you've fixed an embedding of $K$ in $\mathbb C$.
Remark that $ \mathcal O_K \subset \mathbb C$ is an additive subgroup of the complex numbers containing $\mathbb Z$. 
Such a subgroup is discrete if and only if it equals $\mathbb Z$ or a lattice of the form $\mathbb Z+\alpha \mathbb Z$, which happens if and only if $K=\mathbb Q$ or $K$ is an imaginary quadratic field.
Otherwise, $\mathcal O_K\subset \mathbb C$ is not discrete and $0$ is an accumulation point of it, so the unit disc contains infinitely many elements of $\mathcal O_K$. 
A: If $K=\Bbb Q(\zeta_n)$ is cyclotomic, then unless $n=3$ you have

$$\text{rk}_{\Bbb Z}(\mathcal{O}_K^\times)={1\over 2} [K:\Bbb Q] - 1 = {1\over 2}\phi(n) - 1 > 0$$

hence there is a unit of infinite order. Take any such unit, $\varepsilon$. Then either $\varepsilon\in \Delta$ or $\varepsilon^{-1}\in \Delta$ where $\Delta$ is the (interior of the) unit disk. Then either all $\varepsilon^n\in\Delta$ or $\varepsilon^{-n}\in\Delta$. For $K=\Bbb Q(\zeta_3)$ There are none since--unless $a=b=0$, we have

$$\min_{a,b\in\Bbb Z,\; a^2+b^2>0}|a+b\zeta_3|= a^2+b^2-ab\ge \max\{|a|,|b|\}\ge 1$$



*

*The first inequality comes from using $a^2+b^2>0$ and the AM-GM inequality, in the case both are positive you can just use $|ab|\ge \max\{|a|,|b|\}$ since both absolute values are at least $1$ and in the case one is zero, you get it directly by noting only one square remains.


So there is no way to be a non-zero algebraic integer inside the unit disk.
