I am currently solving as many exercises as possible from General linear groups and special linear groups because I am not good in this part of algebra.
The question here is to construct one such monomorphism and I have also given as a hint that the field $\mathbb{F}_4$ can be written as $\{0,1,a,a^2\}$ where $a+1=a^2$ and $y+y=0$ for all $y\in\mathbb{F}_4$.
Is there any way to construct such a monomorphism?
This part that it has to correspond to the inclusion of $A_3$ in $S_3$ troubles me a lot.
I don't know how to start even because I was more used to showing that something is a homomorphism or an automorphism, but now when I have to think about how to construct something like this it gets a bit too difficult.
Hint: First ignore the hint, and consider the basis $1,a$ in $F_4=\{0,1,a,a^2\}$ over $F_2=\{0,1\}$, and calculate the matrices of multiplications by $1,a,a^2$.
Finally, find an isomorphism $GL_2(F_2)\cong S_3$ and verify that the image of the above embedding corresponds to $A_3$ to justify the 'hint'.