Thread: algebraic numbers View Single Post
2020-03-04, 03:03   #4
Dr Sardonicus

Feb 2017
Nowhere

4,451 Posts

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 $A_1\zeta^1 + A_2\zeta^2 + A_3\zeta^3 +A_4\zeta^4...A_{q-1} + \zeta^{q-1}$ and the expression is unique because the cyclotomic polynomial of degree q-1 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 q-1. In general, what you need is the minimum polynomial.

In Pari-GP, you can ask for minpoly(Mod(f, polcyclo(q))). This is the minimum polynomial, and its degree is the degree you want.

Last fiddled with by Dr Sardonicus on 2020-03-04 at 03:05