Are there any fields with a matrix representation other than $\mathbb{C}$? 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?
 A: Yes.
Let $K$ be a field and $f(x)\in K[x]$ an irreducible polynomial. Then we may construct a matrix with entries in $K$ with characteristic polynomial $f$. To do this, we use the companion matrix or, more generally, we take a basis of the field $K[x]/(f)$ (as a vectorspace over $K$) and construct the matrix $M$ associated to the linear map of multiplication by the image of $x$, call it $\bar{x}$, with respect to this basis. Then $M$ has, as characteristic polynomial, $f(x)$, and by the Cayley-Hamilton theorem $M$ satisfies $f$.
Now we define the map $K[x]/(f)\rightarrow M_n(K)$ where $n=\deg(f)$ by $\bar{x}\rightarrow M$ and elements of $K$ map to the diagonal matrices. This is a matrix representation of any finite extension of $K$.
A: 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).
A: Yes, you can represent $\mathbb F_4$, the field with 4 elements, as the following matrices over $\mathbb F_2$, the field with two elements:
$$\left(
\begin{array}{cc}
 0 & 1 \\
 1 & 1
\end{array}
\right),
\left(
\begin{array}{cc}
 1 & 1 \\
 1 & 0
\end{array}
\right),
\left(
\begin{array}{cc}
 1 & 0 \\
 0 & 1
\end{array}
\right),
\left(
\begin{array}{cc}
 0 & 0 \\
 0 & 0
\end{array}
\right).
$$
You can note that the first matrix is the companion matrix to the irreducible (over $\mathbb F_2$) polynomial $x^2 + x + 1$, and that $\mathbb F_4$ can be constructed as
$$\frac{\mathbb F_2[x]}{\langle x^2 + x + 1 \rangle}.$$
For more information, you can google "matrix representations of finite fields".
