Let $t$ a transcendental element on $\mathbb{F}_2$. Prove that $f(x)=x^3-t^3$ is irreducible in $\mathbb{F}_2(t^3)[x]$.
If $f$ is reducible, then we have a solution of $x^3-t^3$, we can call it $\alpha$. So $\alpha \in \mathbb{F}_2(t^3)$ and then $\mathbb{F}_2(t^3)= \mathbb{F}_2(\alpha) $, but after this? How can I conclude the proof to get the desired outcome?
Thanks in advance for any help.