Suppose $\mathbb{F}_{p^n}$ is a finite field and $\mathbb{F}^*_{p^n}$ is the set of all non zero elements in $\mathbb{F}_{p^n}$. The set $\mathbb{F}^*_{p^n}$ is a cyclic multiplicative group. In several references such as this book and this article, it is states that there is an isomorphism between $\mathbb{F}^*_{p^n}$ and $\{\mathbf{A}^j : j \in \mathbb{Z}\}$, where $\mathbf{A}$ is an $n \times n$ matrix over $\mathbb{F}_{p}$ whose characteristic polynomial is primitive polynomial in $\mathbb{F}_{p^n}=\mathbb{F}_p[x]/(f(x)) $.
For example, there is an isomorphism between $\mathbb{F}^*_{8} = (\mathbb{F}_2[x]/(x^3+x^2+1))^* $ and $\{\mathbf{A}^j : j \in \mathbb{Z}\}$ for $\mathbf{A} =$ \begin{bmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&1 \end{bmatrix}
To construct such matrix, the idea is to find a primitive polynomial $p(x)$ in $\mathbb{F}_{p^n}=\mathbb{F}_p[x]/(f(x)) $ (which is also irreducible over $\mathbb{F}_p$) and construct a companion matrix $\mathbf{A}$ from $p(x)$.
For example the, in $\mathbb{F}_{2^3}=\mathbb{F}_2[x]/(x^3+x^2+1) $, we can take $p(x)=x^3+x^2+1$ as our primitive polynomial and construct the (companion) matrix $\mathbf{A}$ as explained above.
The question is, how to complete the formal proof of this proposition? (This problem is related to this one, which I posted previously).