Primitive element of a finite field

Definition

Let $F$ be a finite field of order $q$. An element $\alpha \in F$ is a primitive element of $F$ if $\alpha$ generates the multiplicative group $F^\times$. That is, $F^\times = \{ \alpha, \alpha^2, \dots, \alpha^{q-1} = 1 \}$.

Remarks

Primitive elements exist, because $F^\times$ is cyclic. There are exactly $\varphi(q - 1)$ of them, where $\varphi$ is Euler's totient function. The element $2$ is a primitive element of $\mathbb{F}_3$ and of $\mathbb{F}_5$, but not of $\mathbb{F}_7$. The element $3$ is a primitive element of $\mathbb{F}_7$. A primitive element $\alpha$ of $F$ generates $F$ as a simple extension of its prime subfield. The converse is false in general.

Explanations

L1 human

A primitive element is one whose powers give every nonzero element of the field. It works like a base for logarithms in the finite field.