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 $x$?

Partial answers are welcome too.


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