Show the principal ideal $(x+y\omega)$ has no prime ideal factor in common with other principal ideal $(x+y\omega ^j)$ for $j=0$ or $j=2,...,p-1$

Question

Show that the principal ideal $(x+y\omega)$ in $\mathbb Z[\omega]$ has no prime ideal factor in common with any of the other principal ideals on the left side of the equation $$(x+y)(x+y\omega)(x+y\omega^2)\cdots(x+y\omega ^{p-1})=x^p+y^p,$$ where $p$ is an odd prime and $\omega=e^{2\pi i/p}$.

I came across this problem in a book called $\textit{Number Field}$. I think this problem may use the fact that ideals factor uniquely (up to order) into prime ideals. However, I don't know how to use it.