Let $ k=\mathbb{C}(t)$ be the field of rational functions in one variable. Find the Galois group over $ k$ of the polynomial $ f(x)=x^3+x+t$ .

*My approach:* I’ve made some progress on this problem. Obviously, $ f(x)$ is irreducible over $ k$ . Otherwise, it should have a root in $ \mathbb{C}[t]$ and this root must divide $ t$ and hence has form $ c_0$ or $ c_ot$ but none of these will be a root of $ f(x)$ .

Let $ K$ its splitting field over $ k$ then I showed that $ K/k$ is normal and separable and hence it Galois extension.

Let’s calculate the discriminant of this polynomial. It is equal to $ $ \Delta_f=-4-27t^2=i^2(4+27t^2).$ $

But how to show that $ 4+27t^2$ is not square in $ \mathbb{C}(t)$ ?

Suppose it is square then $ $ 4+27t^2=\left(\dfrac{a_{n+1}t^{n+1}+\dots+a_1t+a_0}{b_nt^n+\dots+b_1t+b_0}\right)^2$ $ where $ a_i,b_i \in \mathbb{C}$ .

And as you see I do not know what should I do further?