Properties of a certain divisor of the $m^{rm th}$ root of unity in the $m^{rm th}$ cyclotomic ring.

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 $gin Z[xi_m]$, 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!