How do you calculate the kernel of the substitution homorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}[e^{i2\pi /r}]$?

Question

I can do that for "easier" substitutions like $x \mapsto i$, but I don't know enough about primitive $rth$ roots yet. Does this require pieces of Galois theory? $r$ is a positive integer. I think if we let $(a,r) = 1, a\neq 1$, then $\varphi : \mathbb{Z}[x] \rightarrow \mathbb{Z}[e^{i2\pi a/r}]$ (subst. hom) has kernel $I = (x^r - 1)$.

Answer 1

You will find that the kernel is generated by the cyclotomic polynomial $\phi_r(x)$.