I am reading a "proof" of Fermat's last theorem which uses the assumption that $\mathbb{Z}[\xi_p]$ is a UFD (which is actually not true in general). This proof uses the following factorization in $\mathbb{Z}[\xi_p]$ (when $p$ is odd): $$x^p+y^p=\displaystyle{\prod_{k=0}^{p-1} (x+\xi_p^ky)}$$ where $x, y$ are coprime integers not divisible by the prime $p$ and $\xi_p$ denotes a primitive $p$-th root of unity.
Could someone point out why this equality holds? Or maybe suggest some resource where a proof can be found? Thanks.
$-\xi_p^k$ is a root of $t^p+1$ for every $k=0,1,...,p-1$ so $t^p+1=\prod_{i=0}^k (t+\xi_p^k)$. Let's substitute $t=x/y$. Then multiplying both sides by $y^p$ gives the equality you want to prove.
This method, I guess, is inspired by homogenization. For example, given a polynomial $f(x)$, $y^df(x/y)$ is its homogenization, where $d$ is the degree of $f$. The equality you want to show is a factorization of a homogeneous polynomial. Homogenization preserves factorization so we can just consider $x^p+1$ whose homogenization is exactly $x^p+y^p$.