Field where $e^2 + f^2 = -1$ implies existance of matrix $A$ with $A^2 = -E_2$?

Question

Suppose we have a field $K$ with a solution for $e^2 + f^2 = -1$. Is there a matrix $A \in K^{2\times 2}$ with $$ A^2 = \begin{pmatrix}-1 & 0 \\ 0 & -1 \end{pmatrix}? $$

Answer 1

Try first with a symmetric matrix, because $-E_2$ is symmetric. Consider the equation $$ \begin{bmatrix} a & b \\ b & c \end{bmatrix}^2= \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} $$ that becomes \begin{cases} a^2+b^2=-1 \\[4px] ab+bc=0 \\[4px] b^2+c^2=-1 \end{cases} The second equation yields $c=-a$ or $b=0$; discarding the latter case which would imply $a^2=-1$, which we know nothing about, we get $$ a^2+b^2=-1 $$ Thus choosing $a=e$, $b=f$ and $c=-e$ we are done.

Answer 2

$$ \left( \begin{array}{rr} e & f \\ f & -e \end{array} \right) $$