I am an analyst who is new to algebraic number theory. I am learning on Cyclotomic polynomials and came across this problem, which I am having hard time solving.
Let $\xi$ be an $m^{\rm th}$ complex root of unity. For each prime divisor $p$ of $m$, let $\xi_p := {\xi_m}^{\frac{m}{p}}$. Let $$g = \Pi_{prime \space p | m}(1-\xi_p)$$
Let $Z$ denote the ring of integers. Show that $g\in Z[\xi_m]$, divides $m$ in $Z[\xi_m]$ if $m$ is odd and $\frac m2$ if $m$ is even. Furthermore, show that $g$ is co-prime with every integer prime except the odd prime divisors of $m$.
Thanks!