Can any Laurent polynomial symmetric under q->1/q and whose roots are all roots of unity be written as a ratio of products of quantum numbers?

Quantum dimensions are quantum integer fractions (or so I heard). Example: $G_2(\lambda_2)$ (technically it should read q^{1/6}, I know...)

$q^{-10}+q^{-8}+q^{-2}+1+q^2+q^8+q^{10}$ = $q_{12}*q_7*q_2/q_6/q_4$

where $q_n$ is shorthand for $(q^n-q^{-n})/(q-1/q)$.

Obviously the left hand side must be symmetric under q->1/q and all roots are roots of unity. Are these conditions already sufficient that a given Laurent polynome can be converted into a quantum integer fraction? Is there even a constructive algorithm? (BTW, I believe the result should be unique since each quantum integer $q_n$ introduces a new root of unity.) In the example the roots are $(-1)^{m/7}$ and $(-1)^{m/12}$ so I might begin with $l.h.s./q_{12}/q_7$, compute the roots of that etc., and this algorithm might already work.

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If coeffieicnets are in Z, then yes. Then q^N*Laurent poly divides q^M-1 so is a product of cyclotomic polynomials, hence can be written in the given form. – Peter McNamara Aug 21 '11 at 20:59