Quote:
Originally Posted by wildrabbitt
The book I'm reading says that given a qth root of unity \[\zeta\], every polyonomial in \[\zeta\] can be expressed as
\[A_1\zeta^1 + A_2\zeta^2 + A_3\zeta^3 +A_4\zeta^4...A_{q1} + \zeta^{q1}\]
and the expression is unique because the cyclotomic polynomial of degree q1 of which \[\zeta\] is a zero is irreducible over the rational field so \[\zeta\] can't be a root of a polynomial of lower degree with integral coefficients.
I know the proof that cyclotomic polynomials are irreducible but I don't get why..... (don't even know what I'm unclear about).
I'm lost. I asked the question I asked orignally because I thought it might help me understand.

OK, I'm assuming q is prime, and you want the degree of the number represented by some polynomial f (your polynomial, with zeta replaced by x, considered modulo the cyclotomic polynomial).
Of course the degree will always divide q1. In general, what you need is the minimum polynomial.
In PariGP, you can ask for minpoly(Mod(f, polcyclo(q))). This is the minimum polynomial, and its degree is the degree you want.