evaluating norm of sum of roots of unity let $l_1,...,l_n$ be roots of unity.
I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$.
how can I do to prove this?

I don't know why you might think the field norm would be at most $n$ in absolute value. For example, if $x=e^{2\pi i/11}$ then I calculate that the norm from ${\bf Q}(x)$ to $\bf Q$ of $x+x^2+x^4$ is 23. 
