Consider the field extension $\mathbb Q(\zeta\_3,\sqrt[3]2,\zeta\_8)/\mathbb Q(\zeta\_3)$, with intermediate fields $\mathbb Q(\zeta\_3,\sqrt[3]2)$ and $\mathbb Q(\zeta\_3,\zeta\_8)$.

Denote 

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> $L:=\mathbb Q(\zeta\_3,\sqrt[3]2,\zeta\_8)$, $K=\mathbb Q(\zeta\_3)$, $M\_1=\mathbb Q(\zeta\_3,\sqrt[3]2)$, $M\_2=\mathbb Q(\zeta\_3,\zeta\_8)$
> 
> 
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Now I know that the extension $L/K$ is galois. I think that all other subextensions are galois too (?)

How do I see from this, that the Galois group of $L/K$ is abelian?