What does $\mathrm{M}_{2}(\mathbb{F}_{7})$ mean in terms of matrix fields

Question

I just need to know what exactly this means

Answer 1

If $n$ is any positive integer, and $R$ is any ring, I would take the notation $\mathrm{M}_n(R)$ to mean the ring of $n\times n$ matrices with entries from $R$, $$\mathrm{M}_n(R)=\left\{\;\;\begin{bmatrix} a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{n1} & \cdots & a_{nn} \end{bmatrix} \;\;\middle\vert\;\;a_{ij}\in R\right\}$$

Thus $\mathrm{M}_2(\mathbb{F}_7)$ would refer to the collection of all $2\times 2$ matrices whose entries are elements of the field $\mathbb{F}_7$ (which is a synonym for $\mathbb{Z}/7\mathbb{Z}$). The ring $\mathrm{M}_2(\mathbb{F}_7)$ has $7^{2\times 2}=2401$ elements.

However, the word "matrix field" doesn't really make sense. The ring $\mathrm{M}_n(R)$ is not a field unless $n=1$ and $R$ is a field.