evaluating norm of sum of roots of unity

Question

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?

Answer 1

You can use the triangle inequality for that, i.e. $|a+b|\leq |a| + |b|$, then the result follows immediately by induction. (BTW: That the norm is larger $-n$ should be obvious, since a norm is always non-negative).

Answer 2

We know that $l_k = e^{(\frac{2\pi i (k-1)}{n})}$, for $k=1,...n$. Then: $$a = \sum_{k=1}^n e^{(\frac{2\pi i (k-1)}{n})}$$ We can evaluate the norm of $a$ and use the triangular inequality: $$|a| = \left|\sum_{k=1}^n e^{(\frac{2\pi i (k-1)}{n})}\right| \leq \sum_{k=1}^n \left|e^{(\frac{2\pi i (k-1)}{n})}\right| = \sum_{k=1}^n 1 = n$$ Then $|a| \leq n$. Anyway, the norm is always a positive (or null) number, so it is always satisfied that it is not smaller than $-n$.