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?
Best Answer
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