Cyclotomic fields - displaying powers of zeta
Is there any way how to avoid expressing $\zeta^{n-1}$ in terms of lower powers of $\zeta$ when working with cyclotomic fields?
K.<zeta> = CyclotomicField(3)
zeta^5
-zeta - 1
I would like to show $\zeta^2$ in this case.
I know that I can do this instead:
K.<zeta> = QQ[]
zeta^5 % (zeta^3 - 1)
zeta^2
But the problem is that eventually, I want to work with a polynomial ring over K and perform a factorization there, and that is not possible in QQ[].<x,y>.
zeta^5 % (zeta^3 - 1)is not the same as computingzeta^5inCyclotomicField(3)as the latter is done modulo cyclotomic polynomialzeta^2 + zeta + 1. So it's unclear what you want.Basically I want a structure where I have both
zeta^2(or at least for display) and ability to factor in the polyring.Then work modulo
zeta^3 - 1while you don't do factoring; and when you do, embed the argument into the cyclotomic field first.