Is well known that the field $\mathbb C$ of complex numbers can be embedded in the ring $M_2(\mathbb R)$ of matrices of order two over de reals. In fact, $\varphi :\mathbb C\longrightarrow M_2(\mathbb R),$ defined by $$\varphi(a+ib)=\left(\begin{array}{cc}a&b\\-b&a\end{array}\right),$$ is an embedding.
Note that $\mathbb C=\mathbb R(i)$, that is, the field $\mathbb C$ is a simple algebaric extension of degree 2 of the reals $\mathbb R$. My question is:
Let $\mathbb F$ be a field and $\mathbb F(a) $ a simple algebaric extension of degree $n$, can the field $\mathbb F(a) $ be embedded in the matrix ring $M_n(\mathbb F)$?
Let $F$ be a field and let $K$ be an extension of $F$ of degree $n$.
Then $\mu: K \to End_F(K)$ given by $\mu(a)(x)=ax$ is an injective ring homomorphism.
Choosing a basis for $K$ over $F$ gives a ring isomorphism $End_F(K) \cong M_n(F)$ and so an embedding $K \to M_n(F)$.