Denote $F=\mathbb{C}(t)$ and consider $p(x)=x^{3}-t\in F[x]$
Is it true that $p$ is irreducible over $F$ ?
My thoughts:
I think that since it is not true that $t^{2}\mid t$ (I don't know how to type not divide) and since $\mathbb{C}(t)/\langle t\rangle\cong\mathbb{C}$ is an integral domain then by Eisenstein the claim follows
Am I correct ? I am not sure that indeed $t$ is prime and that I can apply Eisenstein in my case