The motivation of the question is that I try to test when a positive real number is not an algebraic integer. 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 $d \neq 2\cos(\pi/n), \ n=3,4,5,\cdots$