I am looking to find the Galois group of $x^3-x+t$ over $\mathbb{C}(t)$, the field of rational functions with complex coefficients. I have shown that the automorphisms of the rational function field $F(t)$ for fixed $F$ are precisely the fractional linear transformations that is $t \rightarrow \frac{at +b}{ct+d}$ for $a,b,c,d \in \mathbb{C}$. Is this useful? Also is there anyway to factor $x^3-x+t$ nicely?
I slept on this for a little bit and developed an idea to show this. I used Cardano's method to explicitly solve for the roots of this polynomial and show that there exist no linear factors in $\mathbb{C}(t)$ and $f(x)$ is therefore irreducible. This is because the polynomial is cubic, and if there are no linear factors then there cannot be any quadratic factors. Thus, you have to adjoin some root let's call it $\theta$ to $\mathbb{C}(t)$. The degree of this field over $\mathbb{C}(t)$ is a Galois extension and must have degree 3. The only group with order 3 is $\mathbb{Z}_3$, which implies this is the Galois group.