What are the necessary conditions for a real number to be a cyclotomic integers？ 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.
 A: I suspect you might be looking for the following fact:

If $x$ is a cyclotomic integer, and $p$ a prime does not divide the discriminant, then the minimal polynomial of $x$ factors modulo $p$ into irreducible components all of the same degree.

See for example Theorem 4.6 in Elementary and analytic theory of algebraic numbers by Władysław Narkiewicz.
In practice, this very effectively detects algebraic integers which are not cyclotomic, as used for example in Algorithm 3.6 of David Penneys and James E. Tener, Subfactors of index less than 5, Part 4: Vines, Internat. J. Math. 23 (2012), no. 3, 1250017, 18.
A: Since you are looking for restrictions on quantum dimensions of objects in unitary fusion categories, you also want your cyclotomic integers to be totally real, as they are Frobenius-Perron eigenvalues of finite graphs. The recent article of Calegari-Guo http://arxiv.org/pdf/1502.00035v1.pdf (see Proposition 4.3 in particular) extends the results in the Calegari-Morrison-Snyder article referenced above by @David Speyer. 
