A principal nth root ω of unity is a root satisfying the equations ω^n = 1 and sum_(i = 0)^(n - 1) ω^(i j) = 0 for j = 1, 2, ..., n. Therefore, every primitive root of unity of fixed degree n over a field is a principal root of unity, although this is not in general true over rings. Informally, the term "principal root" is often used to refer to the root of unity having smallest positive complex argument.
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