I wish to find out if $t^{2}$ is a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$ so I can justify the use of Eisenstein on the polynomial $x^{2}-t^{2}\in\mathbb{F}_{2}(t^{2},s^{2})[x]$
I believe that it does since $\mathbb{F}_{2}(t^{2},s^{2})/\langle t^{2}\rangle\cong\mathbb{F}_{2}(s^{2})$ is an integral domain (since it is a field)
Is my argument correct and I may use Eisenstein ? (I already noted that $t^{4}$ does not divide $t^{2}$)