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?

It can be as small as you want.  Take $p=5$, so $\omega^2+\omega^3=-\frac 12(1+\sqrt 5)$.  As this is irrational, by the equidistribution theorem we can find integers $a,b$ so that $a-b\frac 12(1+\sqrt 5)$ is as small as we want.
