Primitive root of unity

Let $F$ be a field and let $\overline{F}$ be an algebraic closure . An element $\zeta\in\overline{F}$ is an $n$-th root of unity if $\zeta^n=1$. It is a primitive $n$-th root of unity if its multiplicative order is exactly $n$, i.e.

Equivalently, $\zeta$ is primitive of order $n$ if and only if it is a root of the cyclotomic polynomial $\Phi_n(x)$.

When $\mathrm{char}(F)\nmid n$, the polynomial $x^n-1$ has distinct roots in $\overline{F}$ (a separability phenomenon; compare distinct-root criterion ), and the $n$-th roots of unity form a cyclic subgroup of $\overline{F}^\times$. Adjoining a primitive $n$-th root produces the cyclotomic extension $F(\zeta)/F$.

Examples

1. Complex numbers. In $\mathbb{C}$, $\zeta_n=e^{2\pi i/n}$ is a primitive $n$-th root of unity, and all primitive $n$-th roots are $\zeta_n^k$ with $\gcd(k,n)=1$.

2. Small orders. Primitive 3rd roots of unity are the two nontrivial roots of $x^2+x+1$, namely $e^{2\pi i/3}$ and $e^{4\pi i/3}$. Primitive 4th roots of unity are $\pm i$.

3. Finite fields. If $F=\mathbb{F}_q$ is a finite field , then $F^\times$ is cyclic of order $q-1$ (see cyclic multiplicative group ). Hence $F$ contains a primitive $n$-th root of unity exactly when $n\mid(q-1)$.