Prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$.

Question

$K/F$ is a field extension of finite degree $n$. How can I prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$, so the ring of $n\times n$ matrix over $F$ contains an isomorphic copy of every extension of $F$ of degree $\leq n$?

Answer 1

The standard way of doing this:

Given any $\alpha \in K$ the map $T_{\alpha}: K \to K$, $x \mapsto \alpha x$ is $F$-linear. Then, fix a basis $B$ of $K$ over $F$. In $B$, $T_{\alpha}$ has a matrix representation.

Another (more clever) idea, but it involves more machinery:

$K/F$ is a finite extension, hence algebraic (try to prove this yourself if you were unaware). Hence for all $\alpha \in K$ we have a minimal polynomial $m_{\alpha}$ for $\alpha$ over $F$. There's a way to associate a matrix to such a polynomial.