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**ThePerfectHacker** If $\displaystyle p(x)$ is irreducible and $\displaystyle \mathbb{F}_q = \text{GF}(q)$ then $\displaystyle \mathbb{F}_q[x]/(p(x))$ is a field. Furthermore, $\displaystyle \mathbb{F}_q$ can be identified as a subfield of this field by an embedding $\displaystyle a \mapsto a + (p(x))$. And so $\displaystyle \mathbb{F}_q[x]/(p(x))$ can be regarded as an extension field of $\displaystyle \mathbb{F}_q$. In fact it is an extension of degree $\displaystyle m$, do you see why?

And I am not sure about you other question, I do not know what you mean by 'primitive'?