Primitive root of unity

Definition. Let $L$ be a field and let $n\ge 1$. An element $\zeta\in L^\times$ is an n-th root of unity if $\zeta^n=1$. It is primitive if its multiplicative order is exactly $n$, i.e.

Equivalently, $\zeta$ generates the cyclic subgroup of all $n$-th roots of unity in $L^\times$ (compare cyclic subgroups and cyclicity of finite-field units ).

Primitive $n$-th roots are precisely the roots of the cyclotomic polynomial $\Phi_n(x)$.

See also. cyclotomic extension , order divides group order .

Examples.

1. In $\mathbb{C}$, $\zeta_n=e^{2\pi i/n}$ is a primitive $n$-th root of unity.
2. $\zeta_3=\frac{-1+i\sqrt3}{2}$ is a primitive cube root of unity; it satisfies $x^2+x+1=0$.
3. In a finite field $\mathbb{F}_q$, an element of order $n$ exists iff $n\mid (q-1)$, since $\mathbb{F}_q^\times$ is cyclic of order $q-1$.