 Primitive element of a finite field

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.