TITLE: Are there any fields with a matrix representation other than $\mathbb{C}$?
QUESTION [8 upvotes]: There has been a lot of research into matrix groups, and even matrix rings, but the only field that I have heard of that has a matrix representation is $\mathbb{C}$. Are there any other fields with a matrix representation? Also, if so, what is the name of the study of matrix representations for fields?

REPLY [5 votes]: Whenever $K$ is a field and $L\subseteq K$ is a subfield, you can consider $K$ as a vector space over $L$.  If $K$ is finite-dimensional as a vector space over $L$, you can choose a basis and identify $K\cong L^n$.  For every $\alpha\in K$, the map $\mu_\alpha:K\to K$ given by $\mu_\alpha(\beta)=\alpha\beta$ is $L$-linear, and so can be represented as an $n\times n$ matrix with entries in $L$.  In this way, the field $K$ can be identified with a subring of the matrix ring $M_n(L)$.  The standard matrix representation of complex numbers is just a special case of this, when $K=\mathbb{C}$, $L=\mathbb{R}$, and the basis chosen is $\{1,i\}$.
These representations are used all the time in field theory (for instance, to define the norm and trace).  However, I would not say that "matrix representations of fields" are themselves studied very much, mainly because their theory turns out to be very simple.  In particular, it is not difficult to show that the representation above is essentially the only way to represent $K$ by matrices with entries in $L$ (this is essentially just the statement that every finite-dimensional vector space over $K$ has a basis).