Cyclicity of the Multiplicative Group of a Finite Field

Theorem.
If $\mathbb{F}_q$ is a finite field , then the group $\mathbb{F}_q^\times$ of nonzero elements under multiplication is cyclic of order $q-1$. Equivalently, there exists $g\in\mathbb{F}_q^\times$ such that every nonzero element is $g^k$ for some $k$.

This is the same statement as finite-field multiplicative group is cyclic .

Examples.

1. In $\mathbb{F}_5$, $2$ is primitive: $2^1,2^2,2^3,2^4\equiv 2,4,3,1\pmod 5$.
2. In $\mathbb{F}_7$, $3$ is primitive: $3^k$ for $k=1,\dots,6$ runs through all nonzero residues mod $7$.
3. In $\mathbb{F}_8$, $\mathbb{F}_8^\times$ has order $7$ (prime), hence is cyclic; any element $\ne 1$ generates.

Related. cyclic groups , finite field structure .