Let $F$ be a finite field and let $a,b \in F$. Prove that $x^2-a$, $x^2-b$ and $x^2-ab$ can't all be irreducible in $F[x]$ where $F$ is a finite field.
I tried to derive a contradiction by assuming $F(\sqrt{a}) = F(\sqrt{b}) = F(\sqrt{ab})$ where $[F(\sqrt{a}):F] = 2$ and then using theory about vector spaces (in particular, using that (1,$\sqrt{a}$) would be a basis), but could not find anything. Any suggestions?
If $F$ is finite, $(F^*, \times)$ is cyclic generated by $u$, so $a$ is not a square implies $a=u^{2n+1}$, $b$ is not a square implies that $ b=u^{2m+1}$ implies $ab$ is a square.