The motivation of the question is that I try to test when a real number is not an cyclotomic integers. Or more specifically, when a positive real number is not a quantum dimension of a unitary fusion category?

We know that when $1\leq d<2$, $d$ is not a quantum dimension of a unitary fusion category if $d \neq 2\cos(\pi/n), \ n=3,4,5,\cdots$

One possible answer is an efficient algorithm to find an approximation of a number in terms of a cyclotomic integer. Just like there is an efficient algorithm to find an approximation of a real number in terms of a rational number.

totally realcyclotomic integers. With this, it becomes quite interesting. $\endgroup$ – Scott Morrison♦ Oct 28 '15 at 4:35