Finite Field Multiplicative Group is Cyclic

Theorem.
If $\mathbb{F}_q$ is a finite field , then its multiplicative group

is a group that is cyclic of order $q-1$. In particular, there exists a primitive element $g\in \mathbb{F}_q^\times$ such that $\mathbb{F}_q^\times=\langle g\rangle$.

Examples.

1. $\mathbb{F}_5^\times=\{1,2,3,4\}$ is cyclic of order $4$; $2$ is a generator since $2,2^2=4,2^3=3,2^4=1$.
2. $\mathbb{F}_7^\times$ is cyclic of order $6$; $3$ generates because its powers give all nonzero residues mod $7$.
3. $\mathbb{F}_8^\times$ has order $7$, hence is cyclic (every group of prime order is cyclic).

Related. cyclic subgroups , finite fields exist uniquely .