Q:

Cyclotomic Polynomial $\Phi_p(x)$ Expansion

On page 310 of Dummit & Foote, the following expansion is given for prime $p$.
$$\Phi_p(x)=\frac{x^p-1}{x-1}=x^{p-1}+x^{p-2}+\cdots+x+1.$$
This is a very basic question, but how does the expansion come about?

A:

It actually has nothing to do with p being a prime number. If you multiply by $(x-1)$ the right hand side you get :
$$ (x-1)\sum_{k=0}^{p-1} x^k = \sum_{k=0}^{p-1} x^{k+1} - x^k = x^p -1$$
Notice that 1 is a root of the polynomial $x^p-1$ so it is natural to try and find a polynomial P such that $P = \dfrac{x^p-1}{x-1}$. One can also notice that it is simply a geometric sum (the proof of the geometric sum is actually the same as the one I provided!)

