For curiosity reasons, I'm trying to see wether or not it is possible to define square roots of rational numbers using (maybe infinite) matrices, without real numbers, ie. using only integers or rational numbers. I'm guessing it is not possible, but I can't find any proof of this.
For example, one could define a number $n$ using two by two matrices, as $nI = \begin{pmatrix} n & 0 \\ 0 & n \\ \end{pmatrix}$
In which case, a square root candidate could be defined as $\sqrt{n} := \begin{pmatrix} 0 & 1 \\ n & 0 \\ \end{pmatrix}$ because $\begin{pmatrix} 0 & 1 \\ n & 0 \\ \end{pmatrix} \begin{pmatrix} 0 & 1 \\ n & 0 \\ \end{pmatrix} = \begin{pmatrix} n & 0 \\ 0 & n \\ \end{pmatrix} $
The problem with this definition is that $\sqrt{n}\sqrt{p} \neq \sqrt{np}$ :
$\begin{pmatrix} 0 & 1 \\ n & 0 \\ \end{pmatrix} \begin{pmatrix} 0 & 1 \\ p & 0 \\ \end{pmatrix} = \begin{pmatrix} p & 0 \\ 0 & n \\ \end{pmatrix} \neq \begin{pmatrix} 0 & 1 \\ np & 0 \\ \end{pmatrix} $
I'm trying to find a working candidate respecting this condition as much as I'm trying to find a proof that it is not possible, but I find it hard to investigate matrices of higher, possibly infinite dimensions.