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# Roots of Unity

A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

What is the complex solution set of the equation \(z^4-1=0\)?

Which of these is a solution to the equation \(z^6=12\)?

Let \(U_n\) denote the set of all \(n^\text{th}\) roots of unity.

If \(U_a=\{yz\mid y\in U_3,\ z\in U_5\}\), then \(a=\ ?\)

Which of these is a solution to the following equation?

\(x^5+x^4+x^3+x^2+x+1=0\)

*not necessarily* an \(n^\text{th}\) root of unity?

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