Subfields of splitting field

Question

I recently started learnig field and Galois theory and was given this problem:

Given polynomial over Q

$\displaystyle f=2x^{6} \ −\ 4x^{5} \ −\ 3x^{4} \ −\ x^{3} \ +\ 8x^{2} \ +\ 6x\ −\ 6\ \in \ Z[ x] \ $

find splitting field K of f, and determine degree [K : Q].

Then find all subfields M of K satisfying [K : M]=2.

I managed to find splitting field K as $\displaystyle \mathbb{Q}\left( \sqrt[3] 2,\zeta _{3,}\sqrt{3}\right) \ $ and [K : Q] = 12

where $\displaystyle \zeta _{3}$ is 3rd root of unity.

I also think that galois group of f is non abellian.

However when I try to figure out to what group it is precisely I get stuck. If I missunderstood something or made mistake, please let me know. If not, how should I proceed?

Answer 1

Hint: We have $$ f=(x^3 - 2)(2x^2 - 6x + 3)(x + 1). $$

Further references:

Galois group of a reducible polynomial over $\mathbb {Q}$

Galois group of a reducible polynomial