Galois group exercise. $x^6+3$

Question

I'm having some difficulties finding the Galois group of the polynomial $g(x)={x}^{6}+3$ over $\mathbb{Q}$.
Here's what I did :
I observed that the roots of the given polynomial are $\sqrt[{6}]{{3}}\xi_{12}^{k}$ where $\xi_{12}$ is a primitive 12-th root of the unity and $k=1,3,5,7,9,11$. Called $\mathbb{K}$ the splitting field of $g(x)$ over $\mathbb{Q}$, is obvious that $\mathbb{Q}(\sqrt[{6}]{{3}},\xi_{12})=\mathbb{Q}(\sqrt[{6}]{{3}},i)\supseteq\mathbb{K}$ so $6|[\mathbb{K}:\mathbb{Q}]\le12$. But from this point I'm not able to continue rigorously.
Seems to me that $[\mathbb{K}:\mathbb{Q}]=6$ but I'm not sure on how to proof that. Can anyone please help me? Thanks in advance!

Answer 1

First, it can't be true that $ [ \mathbb{K} : \mathbb{Q} ] = 6$ since $[ \mathbb{Q}(\sqrt[6]{3}) : \mathbb{Q} ] = 6$ ($x^6 + 3$ is irreducible by Eisenstein criterion) and $\xi_{12} = \exp (i \pi / 6) = \frac{\sqrt{3}}{2} + \frac{i}{2} \in \mathbb{C}$ and hence $\xi_{12} \notin \mathbb{Q}(\sqrt[6]{3}) \subset \mathbb{R}$. You need to find $[ \mathbb{Q}(\sqrt[6]{3},\xi_{12}) : \mathbb{Q}(\sqrt[6]{3})]$. Note that $$ x^6 + 1 = (x^2 + 1)(x^4 - x^2 + 1)$$ is reducible. Moreover, $\sqrt{3} = (\sqrt[6]{3})^3 \in \mathbb{Q}(\sqrt[6]{3})$ and $i = \sqrt{-1}$ is algebric of degree 2 over this field, thus $\mathbb{K} = \mathbb{Q}(\sqrt[6]{3},i)$ is of degree 12 over $\mathbb{Q}$ since $[ \mathbb{Q}(\sqrt[6]{3},i) : \mathbb{Q}(\sqrt[6]{3}) ] = 2$ ($x^2 + 1$ is still irreducible there).