Let $f(x)\in\Bbb F_p[x]$ be irreducible with degree $n$. Then there is a finite field that contains all roots of $f(x)$, namely $\Bbb F_{p^n}$. However, $f(x)$ may or may not have a root as a generator of the multiplicative group of $\Bbb F_{p^n}$ right? If $f(x)$ contains such root(s), is there a phrase or term to call such $f(x)$? I know from cryptography they call $f(x)$ primitive, and the roots of $f(x)$ which is also a generator of $\Bbb F_{p^n}^*$ primitive elements. However, is it standard in abstract algebra? Moreover, the noun "primitive element" make me think of the "primitive element theorem"(see my PS). Is it related to each other?
PS: the primitive element theorem says when for an extesion$F\le E$, $E=F(u)$ for some $u$. But this $u$ is not related to "generator of multiplicative subgroup", is it?
Well, consider the field $GF(16)$. It corresponds to the quotient field ${\Bbb Z}_2[x]/\langle f(x)\rangle$, where $f(x)$ is an irreducible polynomial of degree $4$ over ${\Bbb Z}_2$. These polynomials are $x^4+x+1$, $x^4+x^3+1$ and $x^4+x^3+x^2+x+1$. The first two of which are primitive and conjugate to each other, while the latter is not primitive. Indeed, it divides $x^5-1$ and so each element is a 5th root of unity.
What you refer to is the notion of primitive polynomial, where each root generates the full cyclic group of the field.