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Show that it is a non-abelian group of order 6, then under the Galois correspondence find the (fixed) subfield corresponding to the subgroup of G of order 3.

I've found the splitting field which is $\mathbb{Q}((-1)^{1/3} \ \sqrt[3]{2})$, and the roots are $\sqrt[3]{2}$, $(-1)^{1/3} \ \sqrt[3]{2}$, and $(-1)^{2/3}\sqrt[3]{2}$.

I know the next step is to work out the permutations of the roots - presumably there will be a 3-cycle and a 2-cycle, and at a guess I'd say the Galois group will be isomorphic to $D_{6}$. However, I'm not sure where to go from there.

With regards to the subfield, again at a guess it seems likely it will be permutations of the cube root of unity. Is this correct? And if so how do I show it?

Edited to correct the splitting field and roots.

Lewy
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  • Hint: what's the degree of a splitting field over $\mathbb{Q}$? – xyzzyz Apr 02 '13 at 11:38
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    Hmm. If by $\root 3\of{-2}$ you mean the real cubic root, then you will find that $\root 3\of2=-\root3\of{-2}$. In other words, you will need the third roots of unity. The square root of $-1$ ($=i$) will not help you this time. – Jyrki Lahtonen Apr 02 '13 at 11:38

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The standard approach is to note first that the splitting field if $E = \Bbb{Q}(\omega, \alpha)$, where $\alpha = \sqrt[3]{2}$, and $\omega$ is a primitive third root of unity.

The roots are $\alpha, \omega \alpha, \omega^{2} \alpha$.

Now $\lvert \Bbb{Q}(\omega) : \Bbb{Q} \rvert = 2$, and letting $L = \Bbb{Q}(\omega)$, we have $\lvert E : L \rvert = \lvert L(\alpha) : L \rvert = 3$.

Thus $\lvert E : \Bbb{Q} \rvert = 6$.

Now the Galois group consists of the six elements you obtain as $$ \alpha \mapsto \omega^{k} \alpha, \qquad \omega \mapsto \omega^{\pm 1}, $$ for $k = 0, 1, 2$.

The subgroup of order $3$ is generated by $\sigma$ which acts as $$ \alpha \mapsto \omega \alpha, \qquad \omega \mapsto \omega. $$ By the Galois correspondence, the fixed field has degree $2$ over $\Bbb{Q}$, and there's an obvious candidate here...

Andreas Caranti
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  • Thank you. It seems this question is a duplicate of the one linked above though, so I'm going to delete it. Perhaps you'd like to copy you answer over there? I'm new to this site, so I'm not sure what happens to answers when I delete a question - I'll wait a couple of hours before doing so. – Lewy Apr 05 '13 at 10:28
  • @Lewy, I don't think there is a point in copying my answer into an oldish question, although my answer is somewhat different from the excellent ones given there. But please keep in mind that you don't have to delete your question - it has been marked as a duplicate, and closed (so no new answers), but it can stand, like many other duplicate questions. – Andreas Caranti Apr 05 '13 at 10:36
  • Ok, in that case I will leave it as I think your answer is worth saving. Thank you for the advice, and the answer. – Lewy Apr 05 '13 at 11:17
  • @Lewy, thanks, and you're welcome. – Andreas Caranti Apr 05 '13 at 11:28