Evaluating minimal polynomial over a field $F$ as a characteristic polynomial for a $F$-linear map.

Question

I'm considering $K/F$ to be an extension of degree $n$. I've shown that for any $a\in K$, the map $\mu_a : K → K$ defined by $\mu_a(x) = ax$ for all $x\in K$, is a linear transformation of the $F$-vector space $K$. I'm now trying to show that $K$ is isomorphic to a subfield of the ring $F^{n×n}$ of $n×n$ matrices with entries in $F$ and that $a$ is a root of the characteristic polynomial of $\mu_a$. Using this procedure I've to find the monic polynomial satisfied by $1+ \sqrt 2+ 4^{\frac{1}{3}}$. How is this done? I think calculating the monic polynomial by squaring and cubing will be too cumbersome in this case. How do I do it using the given process?