Let n be a positive integer and let $\alpha$ , $\beta$ be primitive n-th roots of unity.
a) Show that $\frac{1-\alpha}{1-\beta}$ is an algebraic integer.
b) If $n\geq 6$ is divisible by at least two primes ,show that $1-\alpha$ is a unit in the ring $\mathbb Z[\alpha]$.
For a) I tried with $\alpha =\omega$ and $\beta= \omega^2$ primitive cube rots of unity and got the polynomial $x^2-x+1$. But for nth case the calculation is going tedious.Please help. b)Here we have only to show that $\alpha $ is a nilpotent element.Then we are done.But how to show $\alpha$ is nilpotent?
Thanks in advance!