Principal root of unity

In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element ${\displaystyle \alpha }$ satisfying the equations

In an integral domain, every primitive n-th root of unity is also a principal ${\displaystyle n}$-th root of unity. In any ring, if ${\displaystyle n}$ is a power of ${\displaystyle 2}$, then any ${\displaystyle n/2}$-th root of ${\displaystyle -1}$ is a principal ${\displaystyle n}$-th root of unity.

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