Primitive element of a finite field
= Primitive element of a finite field
{title2=$\operatorname{ord}(\alpha)=q-1$}
A primitive element of $\mathbb F_q$ generates its multiplicative group. This implies $\mathbb F_q=\mathbb F_p(\alpha)$, but an element generating the <field extension> need not generate the multiplicative group. For instance $u\in\mathbb F_3[u]/(u^2+1)$ generates the quadratic extension but has order four, whereas $1+u$ has order eight and is multiplicatively primitive.