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(a) Prove that the Galois group $G$ of $f(x)=(x^{3}-2)(x^{3}-3)$ is isomorphic to the semidirect product of $\mathbb Z_{2}$ with $\mathbb Z_{3}\times\mathbb Z_{3}$, where $\theta$ takes the generator of $\mathbb Z_{2}$ to the automorphism $g\rightarrow g^{-1}$ of $\mathbb Z_{3}\times \mathbb Z_{3}$.

(b) Set $\rho =e^{\frac{2\pi i}{3}}$, so that $L:=\mathbb Q[\rho,\sqrt[3]{2},\sqrt[3]{3}]$ is a splitting field of $f(X)$, describe explititly all the subfields of $L$ that containing $\rho$.

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What have you tried ? –  Belgi Dec 29 '12 at 13:39
    
any automorphism $\phi$ takes $\rho$ to $\rho$ or $\rho^{2}$, and takes $\sqrt[3]{2}$ to one of the three othe roots and the same for $\sqrt[3]{3}$, so the order of the group is 18 –  user53800 Dec 29 '12 at 13:42
    
also I can show that the three generators are respectively keeps two of $\rho, \sqrt[3]{2},\sqrt[3]{3}$ stable and takes one of them to $\rho$ times itself, thus, I have find the three generators of the three subgroup namely$Z_{2},Z_{3},Z_{3}$ –  user53800 Dec 29 '12 at 13:53
    
since $Z_{3}\times Z_{3}$ is normal in the Galois group, and the composite of this group with $Z_{2}$ is the whole group, so we can express it in a semidirect product, take the generator of $Z_{2}$ to the generator of the order 2 element of Aut($Z_{3}\times Z_{3}$) –  user53800 Dec 29 '12 at 13:56
    
@Belgi Hi, but I can not figure out how $\theta$ acts, can you help me with this ? –  user53800 Dec 29 '12 at 14:40

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