I came across this homework problem and I'm stumped. This is the third part of the problem, so to give some context, here's what we have so far:
Let $f(x)=x^3-2 \in \mathbb{Q}[x]$. We know that the splitting field is $K=\mathbb{Q}(\sqrt[3]{2},\omega)$, where $\omega$ is the 3rd root of unity. We have $[K:\mathbb{Q}]=6$, which is also the order of the Galois group $G(K,\mathbb{Q})$. Lastly, $G(K,\mathbb{Q})$ is isomorphic to the symmetric group $S_3$.
Here's the part with which I'm having trouble:
For a single nontrivial subgroup $H$ of $S_3$, find $K_H$ (the fixed field of $H$) and check the correspondence given in the fundamental theorem of Galois theory.
I decided to choose $H=\{id, (1,2,3), (1,3,2)\} \subset S_3$. We know that $K_H=\{a\in K\mid \sigma(a)=a \,\,\forall\,\, \sigma\in H\}$.
Let $\sigma \in H$. Then, we have the three following automorphisms
$\sigma_1 = \begin{pmatrix} 1&2&3\\3&1&2\end{pmatrix}$
$\sigma_2 = \begin{pmatrix} 1&2&3\\2&3&1\end{pmatrix}$
$\sigma_3 = \begin{pmatrix} 1&2&3\\1&2&3\end{pmatrix}$
Let $T=\mathbb{Q}(\omega)$ and let $\sigma\in G(K,T)$.
By looking at $\sigma_1$ or $\sigma_2$, we see that none of $r_1,r_2,r_3$ is in the fixed field $K_H$.
For the correspondence, I'm not exactly sure what the question is asking me to do. The fundamental theorem of Galois theory states that for $\mathbb{Q}\subset T \subset K$, we associate to $T$ the subgroup $G(K,T)\subset G(K, \mathbb{Q})$. Conversely, for $H\subset G(K,\mathbb{Q})$, associate to $H$ the fixed field $K_H$.
Under this correspondence, we have:
(1) $T=K_{G(K,T)}$
(2) $H=G(K,K_H)$
(3) $[K:T]=o(G(K,T))$ and $[T:\mathbb{Q}]=\frac{o(G(K,\mathbb{Q}))}{o(G(K,T))}$
(4) $T$ is a normal extension of $\mathbb{Q}$ iff $G(K,T)$ is a normal subgroup of $G(K,\mathbb{Q})$.
(5) If $T$ is normal over $\mathbb{Q}$, then $G(K,T)$ is isomorphic to the quotient group $G(K,\mathbb{Q})/G(K,T)$
This statement is very long, and I'm not sure if the question is asking me to verify that each of the five conditions holds for my specified $H$ and $T$ in $K$, or if it just wants me to verify that a one-to-one correspondence exists between $H$ and $T$. In either case, I'm not sure how to proceed.
I apologize for the long post. Any guidance at this point is greatly appreciated.