# Showing that a field is isomorphic to a field of vectors

1. Apr 6, 2013

### QIsReluctant

1. The problem statement, all variables and given/known data
Let $K / F$ be a field extension of degree $n$.

For any $a \in K$ prove that the map defined by $\sigma_a(x) = ax$ is a linear map over the vector space $K / F$. This part I understand.

Show that $K$ is isomorphic to the subring $F^{n x n}$ of $n x n$ matrices with entries in $F$.

3. The attempt at a solution
I understand that linear map from a field extension to itself = an $n x n$ matrix. The trick is showing that the operations of addition and multiplication are compatible, and I can't seem to do that without a lot of tedious, convoluted calculations, especially since there's not a good closed form for the transformation matrix that would apply to all bases (as far as I know). Is there a better way?