sage:  K.<p,d,e,N> = FractionField(PolynomialRing(QQ,4,'pdeN'))
sage: R.<x> = K[]
sage: a = x^3-x^-3
sage: b = x^5-x^-5
sage: c = x^8-x^-8
sage: X = p*a + d*b + e*c
sage: f = x^16 *(X^2- N*b*c)

and Sage does not answer.  It just hangs and I have to kill the session.
If it would answer I would like to continue with 

F = R(f)
psi = cyclotomic_polynomial(30)
g = F.quo_rem(psi)[1]
g

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