I cannot seem to get this proof right, which is to show that roots common to $z^{m}-1=0$ and $z^{n}-1=0$ where $\gcd(m,n)=d$ is given only by all the roots of the equation $z^{d}-1=0$.
For starters, if we let $m=d\lambda_{1}$ and $n=d\lambda_{2}$, where $\lambda_{1}, \lambda_{2}\in\mathbb{Z}^{+}$, then $z^{d\lambda_{1}}=1$ holds, and so does $z^{d\lambda_{2}}=1$. I do not know how to formalize this. Any hints are appreciated. Thanks.
