The number of algebraic integer within the unit disk

Question

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

1. whether there are only finitely many such $\alpha$ whose absolute value is less than 1.
2. If so, is there any explicit bound?
3. 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 $n$?

Answer 1

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 $\mathcal O_K$.

Answer 2

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.