## Galois group over the field $\mathbb{C}(t)$

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?