MHB Challenge problem #5 [Olinguito]

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The discussion focuses on the definition of the operation A circ B for nonempty sets of complex numbers and the recursive definition of A raised to the power of n. It establishes that for a fixed integer n and any positive integer r, the set of nth roots of unity, denoted as ζ_n, remains unchanged under the defined operation for any r. Consequently, it concludes that ζ_n^{[r]} equals ζ_n^{[1]} for all r. The sum of the elements in ζ_n, which are the roots of the polynomial z^n - 1, is determined to be zero. Thus, the final result is that the sum of all elements in ζ_n^{[r]} is also zero.
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If $A$ and $B$ are nonempty sets of complex numbers, define
$$A\circ B\ =\ \{z_1z_2:z_1\in A,\,z_2\in B\}.$$
Further define $A^{[1]}=A$ and recursively $A^{[n]}=A^{[n-1]}\circ A$ for $n>1$.

Let $\zeta_n=\{z\in\mathbb C:z^n=1\}$. Given a fixed integer $n\geqslant2$ and any positive integer $r$, find the sum of all the elements in $\zeta_n^{[r]}$.
 
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As $\zeta_n^{[1]}$ is closed under multiplication, $\zeta_n^{[r]}\subset\zeta_n^{[1]}$. On the other hand, as $1\in\zeta_n^{[1]}$, $\zeta_n^{[1]}\subset\zeta_n^{[r]}$.
The conclusion is that $\zeta_n^{[r]}=\zeta_n^{[1]}$ for all $r$. The sum of the elements of that set is the sum of the roots of $z^n-1$, which is $0$.
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Nice work. (Yes)

Alternatively, note that
$$A\circ B\ =\ \bigcup_{z\in A}\,\{z\}\circ B$$
and $\{\omega\}\circ\zeta_n=\zeta_n$ where $\omega$ is any $n$th root of unity.
 
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