Galois group of a polynomial over $\mathbb C(t)$

Question

I am learning for my exam tomorrow and I am facing the following task:

Compute the Galois group of $x^3-2tx+t$ over $\mathbb C(t)$

At first I want to show that this polynomial is indeed irreducible. And it is sufficient if I am able to show that there are no roots. But thb I am not able to handle polynomials like this...

Can someone give ma a tip?

Answer 1

Hint: $\mathbb{C}[t] / (t)$ is indeed isomorphic to $\mathbb{C}$, look at the definition: The qoutient are the rest when you divide through the polynomials without constant term.

In case you want to find the homomorphism: Try $\mathbb{C}(t) \rightarrow \mathbb{C}, f(t) \mapsto f(0)$, the kernel is exactly the ideal $(t)$ and the map is surjective. Then it's just the homomorphism theorem.

After this: Do you know how to compute the Discriminant of a polynomial with degree 3?