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I'm a professional mathematician and professor.

This is a very interesting question, and a great motivator for Galois theory, kind of like a Zen koan. (e.g. "What is the sound of one hand clapping?")

But the question is inherently imprecise. As soon as you make a precise question out of it, that question can be answered trivially.

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Generally, the nth roots of 1 form a cyclic group (with complex multiplication, i.e. rotation by multiples of 2pi/n).

One of the roots is 1, choosing either adjacent one as a privileged group generator means choosing whether to draw the same complex plane clockwise or counterclockwise.

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