Factorising $X^n+...+X+1$ in $\mathbb{R}$ How can factorize this polynom in $\mathbb{R}$:
$X^n+...+X+1$
I already try to factorize it in $\mathbb{C}$ but I couldn't find a way to turn to $\mathbb{R}$

We have
$$\sum_{k=0}^n x^k=\frac{x^{n+1}-1}{x-1}$$
hence
$$\sum_{k=0}^n x^k=\prod_{k=1}^{n}\left(x-e^{2ik\pi/{n+1}}\right)$$
so if $n$ is odd say $n=2p+1$ then
$$\sum_{k=0}^{2p+1}=\prod_{k=1}^{2p+1}\left(x-e^{2ik\pi/{2p+2}}\right)=(x+1)\prod_{k=1}^{p}\left(x-e^{2ik\pi/{2p+2}}\right)\prod_{k=p+2}^{2p+2}\left(x-e^{2ik\pi/{2p+2}}\right)\\=(x+1)\prod_{k=1}^{p}\left(x^2-2\cos(2k\pi/2p+2)+1\right)$$
and the case $n$ is even is left for you.
