Let $f(x)=x^3+11 ∈ \mathbb Q[x]$. Let $E=\mathbb Q^f(x)$ be the splitting field of $f(x)$.

Question

Let $f(x)= x^3+11 ∈ \mathbb Q[x]$. Let $E=Q^f(x)$ be the splitting field of $f(x)$.

1. Find $\mathbb Q^f(x)$ and the degree of $\mathbb Q^f(x)$ over $\mathbb Q$ .

2. List all the elements of the Galois group $Gal (\mathbb Q^f(x)/\mathbb Q)$.

3. List all the subgroups of the Galois group.

4. For each subgroup of the Galois group, find its fixfield.

Answer 1

I'll help you get started on the first part. Once you understand that I'll add some more.

The three roots of $f(x)$ are: $$-\sqrt[3]{11}, -\omega\sqrt[3]{11},-\omega^2\sqrt[3]{11}.$$

Where $\omega$ is a primitive cube root of unity. From this try to prove that $E = \mathbb{Q}(\omega,\sqrt[3]{11})$.