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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!

Bart Michels
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Germain
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  • no answer and no hints? – Germain Mar 17 '14 at 18:40
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    For a) write $\alpha = \beta^d$ for some $d$ such that $(d,n)=1$. For b) note that a unit like $\alpha$ will never be nilpotent. For a proof of b) look here http://math.stackexchange.com/questions/216035/why-1-zeta-unit-where-zeta-is-a-primitive-nth-and-n-divisible-by-two-pri – benh Mar 17 '14 at 23:58

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