Can one give an elementary proof of following interesting theorem? (These are from the paper "Finite subgroups of $PGL(2,K)$-Beauville").
(In each statement , characteristic of $K$ is prime to order of group in statement, as mentioned in the paper above. One may modify the statement if possible. )
$PGL(2,K)$ contains $\mathbb{Z}/r$ and $D_r$ iff $K$ contains $\zeta +\zeta^{-1}$ for some premitive $r^{th}$ root of unity $\zeta$.
$PGL(2,K)$ contains $A_4$ and $S_4$ iff $-$ is sum of two squares in $K$.
$PGL(2,K)$ contains $A_5$ iff $-1$ is sum of squares in $K$ and $5$ is a square in $K$