Primitive root of unity

Let $F$ be a field and $n\geq 1$ an integer. An element $\zeta\in F$ is a primitive $n$th root of unity if $\zeta^n = 1$ and $\zeta^m\neq 1$ for $1\leq m<n$, that is, $\zeta$ has order $n$ in $F^\times$.

When is $\zeta$ a primitive $n$th root of unity?