Hello I would like to calculate all subfields of the field in which $f$ splits in linear factors (I just don't know the right english word for this field :) )
$f:=X^6-2$ is irreducible with Eisenstein criterion. Define $a:=2^\frac{1}{6}$ and $\xi:=\exp(\frac{2\pi i}{6})$. Then $$X^6-2=(X-a)\cdot\ldots\cdot(X-a\xi^5).$$
Now $[\mathbb{Q}(a):\mathbb{Q}]=6$, since $f$ is the minimal polynomial of $a$.
and $[\mathbb{Q}(a)(\xi):\mathbb{Q}(a)]=2$, since $a\in\mathbb{R}$, so $X^2-X+1$ is still the minimal polynomial of $\xi$.
Therefore $[\mathbb{Q}(a,\xi):\mathbb{Q}]=12=ord(Gal(f,\mathbb{Q}))$, since $K:=\mathbb{Q}(a,\xi)$ is the field in which $f$ splits in linear factors.
Now calculate the automorphisms:
First there are automorphisms:
$$\phi:\mathbb{Q}(a)\rightarrow\mathbb{Q}(\xi a), \phi|_\mathbb{Q}=id_\mathbb{Q}, \phi(a)=\xi a$$
and the extension
$$\varphi:K\rightarrow K, \varphi|_\mathbb{Q}=id_\mathbb{Q}, \varphi(a)=\xi a, \varphi(\xi)=\xi.$$
There is also an extension with $\overline{\varphi}(\xi)=\overline{\xi}$, since $\xi$ and $\overline{\xi}$ are roots of the minimal polynomial of $X^2-X+1$, but one can get this by using the product of $\varphi$ and the following automorphism
$$\psi:K=\mathbb{Q}(a)(\xi)\rightarrow\mathbb{Q}(a)(\overline{\xi}),\psi|_\mathbb{Q}=id_\mathbb{Q}, \psi(a)=a, \psi(\xi)=\overline{\xi},$$
since $X^2-X+1$ is still the minimalpolynomial of $\xi$ and $\overline{\xi}$ over $\mathbb{Q}(a)$.
So $G:=Gal(f,\mathbb{Q})=\{id,\varphi,\ldots,\varphi^5,\psi,\psi\varphi,\ldots,\psi\varphi^5\}$.
So one gets the subgroups of $G$ of order $2,3,4$ or $6$.
But now I am having problems to calculate the subfield. For example for $\psi\varphi^3$ one has:
$$[K:K^{<\psi\varphi^3>}]=ord(<\psi\varphi^3>)=2$$ and $$[K^{<\psi\varphi^3>}:\mathbb{Q}]=6$$.
($K^{<\psi\varphi^3>}$ denotes the usual fixgroup) Now the usual approach I know would be to find an linear combination $c$ of elements of the fixgroup and show by using the degree formula that $K^{<\psi\varphi^3>}=\mathbb{Q}(c)$ but I cannot find such $c$. So what is the usual approach to solve this?
Thanks for your help!