algebraic conjugate and sum of roots of unity 
In above lemma, why $|a'| \leq 1$ still holds? I didn't see how it relates to "algebraic conjugate of a root of unity is also a root of unity", since $a$ is the sum of unity.
(definition of algebraic conjugate: http://planetmath.org/algebraicconjugates)

We have $\sigma (\sum \epsilon_i/n) = \sum\sigma(\epsilon_i)/n$, since each $|\sigma(\epsilon_i)|=1$, $|a'|=|\sigma(a)|=1$.
