# Computing the sum of inverses of some roots of 1 in a field, given their sum

Fix an algebraically closed field $$F$$.

Let $$\alpha_1,\dotsc,\alpha_n\in F$$ be roots of $$1$$. Let $$x=\alpha_1+\dotsc+\alpha_n$$ and $$y=\alpha_1^{-1}+\dotsc+\alpha_n^{-1}$$.

I was thinking: Given $$n$$ and $$x$$, can we compute $$y$$?

If $$F=\mathbb{C}$$ the answer is positive: $$y$$ is the complex conjugate of $$x$$.

If $$F$$ is the algebraic closure of the field with $$5$$ elements, the answer is negative. Both $$\alpha_1=1,\alpha_2=1$$ and $$\alpha_1=3,\alpha_2=4$$ give $$x=2$$, but the former gives $$y=2$$ while the latter gives $$y=1$$.

So the answer depends on $$F$$.

Question: What are the algebraically closed fields $$F$$ where we can always compute $$y$$ as a function of $$n$$ and $$x$$?