Root of unity modulo n

In mathematics, namely ring theory, a k-th root of unity modulo n for positive integers k, n ≥ 2, is a solution x to the equation (or congruence) ${\displaystyle x^{k}\equiv 1{\pmod {n}}}$. If k is the smallest such exponent for x, then x is called a primitive k-th root of unity modulo n. See modular arithmetic for notation and terminology.

Do not confuse this with a Primitive root modulo n, where the primitive root must generate all units of the residue class ring ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ by exponentiation. That is, there are always roots and primitive roots of unity modulo n for n ≥ 2, but for some n there is no primitive root modulo n. Being a root of unity or a primitive root of unity modulo n always depends on the exponent k and the modulus n, whereas being a primitive root modulo n only depends on the modulus n — the exponent is automatically ${\displaystyle \varphi (n)}$.

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