Minimum absolute values of algebraic integers in cyclotomic fields Let $\omega$ be a primitive $p^{th}$ root of unity with $p$-odd prime. 
Consider $\mathbb{Z}[\omega]$, the ring of integers in cyclotomic field $\mathbb{Q}(\omega)$. I wanted to know how small can be the elements of $\mathbb{Z}[\omega]$, in the sense - 
Is the set $$\{|\alpha|: \alpha\in\mathbb{Z}[\omega], \alpha\neq 0\}$$
bounded below?

If you want something less beefy than the equidistribution theorem, just take $p$ to be whatever you like. You know that $\sqrt{p^*}\in\Bbb Z[\omega]$ where $p^*=(-1)^{p-1/2}p$, and since $p^*\equiv 1\mod 4$ we have ${2j+1+\sqrt{p^*}\over 2}\in\Bbb Z[\omega]$ is an algebraic integer for all $j\in\Bbb Z$. If $p\equiv 1\mod 4$ then let $n<\sqrt{p^*}=\sqrt{p}<n+1$. There are two cases:

$$\alpha = \begin{cases}
{n-\sqrt{p^*}\over 2} & n = 2k+1 \\
{n+1-\sqrt{p^*}\over 2} & n= 2k
\end{cases}$$

both of which give a number, $\alpha$, with absolute value $<1$, in particular its powers converge to $0$, which gives an infinite family of rings with small absolute values.
