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Let $k$ be an algebraically closed field of characteristic $0$ and $\alpha, \beta$ two distincts roots of unity of the same order, i.e both solutions of $x^n-1 = 0$, for some $n$ positive integer.
If $k = \mathbb{C}$, we know there is some $p,q \geq 2$ integers such that $\alpha^p = \beta$ and $\beta^q = \alpha$, does this hold for any field $k$ as defined above?

If the cardinality of $k$ is smaller than the cardinality of $\mathbb{C}$ we could embbed $k$ into $\mathbb{C}$ to find such $p$ and $q$, but in the other case, I'm not sure how to proceed. Am I missing something?

Thanks in advance.

square17
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    See https://math.stackexchange.com/questions/59903/finite-subgroups-of-the-multiplicative-group-of-a-field-are-cyclic – Captain Lama Mar 21 '22 at 17:47
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    If $z$ is a primitive cube root of unity, then both $z$ and $-1$ are $6$-th roots of unity but neither is a power of the other. Do you mean to only consider primitive $n$-th roots of unity? – Ned Mar 21 '22 at 17:54

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