Why is $\alpha$, a primitive pth root of unity, a root of the polynomial $1+t+\ldots +t^{p-1}$?

Question

I'm currently reading a proof that the minimal polynomial of $\alpha$, a primitive pth root of unity, over $\mathbb{Q}$ is $$1+t+\ldots +t^{p-1},$$ yet the author simply states "Clearly $\alpha$ is a zero." in the proof, and I can't seem to prove that bit myself.

Answer 1

Note $(1+t+\cdots + t^{p-1} ) (1-t)= t^p -1$, and that $\alpha$ is a root of $t^p -1$, since it is a pth root of unity, but not of $1-t$, since it is primitive. Then it must be a root of $1+\cdots+t^{p-1}$, as the author says.