I know that the ring
$(AS,+,\cdot)$, where
$$AS := \bigg\{\bigg( \begin{matrix} a & - b \\ b & a \end{matrix} \bigg) \; : \; a,b \in \mathbb{R} \bigg\}$$
and $+$ is the matrix addition and $\cdot$ is the matrix multiplication, is a field and it's isomoprhic to the field of the complex numbers $(\mathbb{C},+,\cdot)$
this is a consequence of the fact that $AS = \mathbb{R}[J]$ where
$$J = \bigg( \begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix} \bigg)$$ is a matrix such that $p_J(t) = t^2 + 1$, which thanks to the Cayley-Hamilton theorem tells me that $J^2 + I = O$ where $I$ is the identity matrix and $O$ is the null matrix. from this it can be seen pretty easily that $AS = \mathbb{R}[J] \cong \mathbb{C}$
I would like to generalize this result, therefore I would like to prove this theorem
Theorem
Let $\mathbb{K}$ be a field, let $\mathbb{F}$ be a simple algebraic extension of $\mathbb{K}$ of degree $n$, then there exists a subring $MR$ of the ring $(\mathbb{K}^{n \times n}, + , \cdot)$ such that $MR$ is a field and $MR \cong \mathbb{F}$
So basically I have to prove that there exists a matrix $A \in \mathbb{K}^{n \times n}$ such it's Characteristic polynomial equals the minimal polynomial of the extension $\mathbb{F}$ of $\mathbb{K}$.
More in general, if $q \in \mathbb{K}[t]$ is a polynomial of degree $n$, there exists a matrix $A \in \mathbb{K}^{n \times n}$ such that $p_A(t) = q(t)$ ? If so, how can I prove it?
For any polynomial $q(t) = t^n + a_{n-1} t^{n-1} + a_{n-2} t^{n-2} + ... + a_1 t + a_0 \in \mathbb K[t]$ of degree $n$, then $$ \mathbf A = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-2} & -a_{n-1} \end{pmatrix} $$ is the matrix which $p_{\mathbf A}(t) = q(t)$.
Furthermore, if $\mathbb F = \mathbb K(\theta)$, and $q$ is the minimal polynomial of $\theta$ over $\mathbb K$, then we have $\mathbb F \cong \mathrm{span}\{\mathbf I, \mathbf A, \mathbf A^2, \cdots, \mathbf A^{n-1}\}$ over $\mathbb K$