Let $\mathbb C$ be the field of complex numbers and $\mathbb R$ the field of real numbers. It is well known that the field $\mathbb C$ can be represented as
$$\mathbb C=\left\{\left(\begin{array}{cc}a&b\\-b&a\end{array}\right)\;:a, b\in\mathbb R\right\}.$$ Now, let $\mathbb F$ be a field and $\mathbb K$ a field extension of $\mathbb F$ of finite degree. Can we represent $\mathbb K$ as a matrix ring over $\mathbb F$?
Any finite-dimensional algebra $A$ over some field ${\mathbb k}$ is isomorphic to a subalgebra of $\text{Mat}_n({\mathbb k})$ for some $n$: for example, choosing a ${\mathbb k}$-base $\{v_1,...,v_n\}$ of $A$, the adjoint representation $A\to\text{End}_{\mathbb k}(A)\cong\text{Mat}_n({\mathbb k})$, $a\mapsto a\cdot -$ gives such a description. Taking for example ${\mathbb k}={\mathbb R}$, $A = {\mathbb C}$ and the ${\mathbb R}$-basis $\{1,i\}$ of ${\mathbb C}$ yields the matrix description of ${\mathbb C}$ over ${\mathbb R}$.