Suppose $ A_{n\times n} $ matrix on $ \Bbb F_{p} $ ($P$ is prime) and if the Characteristic polynomial is irreducible then we can make a finite field that the order of $P^n$ with $ A_{n\times n} $ .
$$ \{ 0, I , A^{}, A^{2},A^{3},...,A^{p^{n}-2} \} $$
Is the Characteristic polynomial of $ A_{n\times n} $ is a Primitive polynomial?
for example:
I wanna show $x^{3} + x^{2} +2$ is a Primitive polynomial on $ \Bbb F_{5} $.
let $A= \begin{pmatrix} 0 & 0 & 3 \\ 1 & 0 & 0 \\ 0 & 1 & 4 \end{pmatrix} $ by the cayley hamilton theorem $A^{3} + A^{2} +2 = 0$.
we can show $ \{ 0, I , A^{}, A^{2},A^{3},...,A^{5^{3}-2=123} \} $ is a filed by $125$ order and every field by $5^3$ isomerism. Can i say $x^{3} + x^{2} +2$ is a Primitive polynomial on $ \Bbb F_{5}$?