Assume $K$ is algebraically closed and $\operatorname{char}(K)=0$. Let $n$ be a positive integer. Assume we have $\lambda_1,..., \lambda_m$ $n$th roots of unity with the property $\lambda_1 +...+\lambda_m =m$. Show that all $\lambda_i =1$.
How can we prove this? Is there any method using Galois theory (acting via elements of the galois group) or is it possible to go back to the case $K=\mathbb{C}$ via some isomorphism and then using the absolute value of complex numberes? Other ideas are also welcome!