Let $F$ be a field of characteristic $\operatorname{char}(F)\ne2$. Prove that the polynomial $f(x)=x^4+tx+t\in F(t)[x]$ is irreduciable, and find its Galois group.
Proving that the polynomial is rather simple, using the generalized Eisenstein criterion, taking the ideal $\mathfrak{p}=(t)\in F[t]$ (as $f$ is monic, it will be irreducible in $F(t)[x]$ from Gauss's lemma).
The discriminant is $\Delta_x=-27t^4+256t^3$, the root of which is not an element of $F(t)$. Can I deduce that $\operatorname{Gal}(f)\cong S_4$?