Primitive root of unity

Let $F$ be a field, let $\overline F$ be an algebraic closure, and let $n\ge1$ . An element $\zeta\in\overline F$ is a primitive $n$ th root of unity if its multiplicative order is exactly $n$ :

\zeta^n=1 \quad\text{and}\quad \zeta^d\ne 1 \text{ for every proper divisor } d\mid n.

Equivalent characterizations

If $\operatorname{char}(F)\nmid n$ , then $\zeta$ is a primitive $n$ th root of unity if and only if it is a root of the cyclotomic polynomial $\Phi_n(x)$ . This characterization can fail after reduction to characteristic dividing $n$ .

Remarks

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

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 $\gcd(k,n)=1$ .

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 $\pm i$ .

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 $n\mid(q-1)$ .