Presentation of the algebraic closure of finite fields via matrices

Question

Sorry if this question is too elementary for MO.

Let $p$ be a prime and $F_p$ a field with $p$ elements and $F_{p^n}$ the field with $p^n$ elements Then we can choose an irreducible factor $f$ of degree n of a suitable cyclotomic polynomial in $F_p [x]$, take the companion matrix $C_f$ of $f$ and then $F_{p^n}$ can be representated as $\{0,C_f,C_f^2,...,C_f^{p^n-1} \}$.

In practise this gives a pretty nice presentation of $F_{p^n}$ via matrices over $F_p$ in my opinion.

For example, look at $F_3$ and $f:=x^2+x+2$. Then $C_f= \begin{pmatrix} 0 & 1 \\ 1 & 2 \end{pmatrix}$ and $F_9= \{ 0, C_f^i | i=1,...,8 \}.$

Question: Is there a nice explicit presentation of the algebraic closure of $F_p$ via matrices over $F_p$ that is discussed somewhere in the literature?

Answer 1

I don't know if this counts as nice or explicit, but I'll take a shot. For a matrix $A\in M_n(\mathbb F_p)$ and $n|m$ or $m=\infty$ denote by $d_m(A)$ the matrix of the form $$\left[\begin{array}{ccc}A&&\\&A&\\&&\ddots\end{array}\right]\in M_m(\mathbb F_p).$$

First choose $f_2\in\mathbb F_p[x]$ monic irreducible of degree 2 and let $C_2=C_{f_2}$ be its companion matrix. Let $\mathbb F_{p^2}\subset M_2(\mathbb F_p)$ be the $\mathbb F_p$-subspace spanned by $I_2,C_2$. Now choose $f_3\in\mathbb F_{p^2}[x]$ monic irreducible of degree 3 and let $C_3=C_{f_2}\in M_3(\mathbb F_{p^2})=M_6(\mathbb F_p)$ be its companion matrix. We take $$\mathbb F_{p^6}=\langle d_6(C_2)^iC_3^j\,:\,0\le i\le 1,\,0\le j\le 2\rangle_{\mathbb F_p}\subset M_\infty(\mathbb F_p)$$ and continue recursively: at each step we have $\mathbb F_{p^{n!}}$ already constructed, choose a monic irreducible $f_{n+1}\in\mathbb F_{p^{n!}}[x]$ of degree $n+1$, let $C_{n+1}=C_{f_{n+1}}\in M_{n+1}(\mathbb F_{p^{n!}})= M_{(n+1)!}(\mathbb F_p)$ be its companion matrix and define $$\mathbb F_{p^{(n+1)!}}=\langle d_{(n+1)!}(C_2)^{i_2}\cdots d_{(n+1)!}(C_{n+1})^{i_{n+1}}\,:\,0\le i_2\le 1,\ldots,0\le i_{n+1}\le n\rangle_{\mathbb F_p}\subset M_{(n+1)!}(\mathbb F_p).$$ Finally take $\overline{\mathbb F_p}=\bigcup_{n=1}^\infty\{d_{\infty}(A):\,A\in\mathbb F_{p^{n!}}\}\subset M_\infty(\mathbb F_p)$