I am really struggling on these consecutive questions on Galois Theory:

Question 1: Suppose $L$ is a normal closure of $\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{5}) $ over $\mathbb{Q}$. Find $\mathrm{Gal}(L/\mathbb{Q})$.

Would this be the cyclic product $\mathbb{Z_2} \times \mathbb{Z_3} $ or $\mathbb{Z_6}$?

Question 2: Apply Galois theory to prove that $\sqrt[3]{3} \not\in L$.

$\sqrt[3]{3}$ clearly does not belong to $\mathbb{Q}$ and cannot be obtained using a composition of typical operations (addition, subtraction, multiplication etc) of $\sqrt[3]{2} $ and $\sqrt[3]{5} $ but how can I show this using Galois theory?

Many thanks for your help :-)


Question 1 :

The possible values by an $L$-automorphism for $\sqrt[3]{2}$ are $\sqrt[3]{2}$, $j\sqrt[3]{2}$ and $j^2\sqrt[3]{2}$. And for $\sqrt[3]{5}$ it is $\sqrt[3]{5}$, $j\sqrt[3]{5}$ and $j^2\sqrt[3]{5}$. But the complex $j$ appears in our expressions, so the closure of $\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{5})$ is $\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{5},j)$. And the possible images of $j$ by an $L$-automorphism are $j$ and $j^2$. So by considering all the combinations possible, there is a total of 18 possible $L$-automorphisms, and 17 of these automorphisms are of order 3, 17 are of order 6, one is of order 2 and there is $\mathrm{Id}$, so $\mathrm{Gal}(L/\mathbb{Q})$ is isomorphic with $S_3 \times \mathbb{Z_3}$.

  • $\begingroup$ I'm assuming $j$ is a primitive cube root of unity? What about the map $j\mapsto j^2$? $\endgroup$ – carmichael561 Apr 8 '16 at 17:00
  • $\begingroup$ $j^2$ is also a primitive root of the unity. Indeed $(j^2)^2=j$ and $(j^2)^3=1$ $\endgroup$ – Jennifer Apr 8 '16 at 17:05
  • $\begingroup$ I'm aware of that. My point is you're missing an automorphism. $\endgroup$ – carmichael561 Apr 8 '16 at 17:06
  • $\begingroup$ I uploaded the table of the automorphisms and I don't think I miss any automorphism. But corect my answer if I am wrong. $\endgroup$ – Jennifer Apr 8 '16 at 17:08
  • $\begingroup$ You're missing the map $j\mapsto j^2$. The Galois closure of $\mathbb{Q}(2^{\frac{1}{3}},5^{\frac{1}{3}})$ is $\mathbb{Q}(2^{\frac{1}{3}},5^{\frac{1}{3}},j)$, which has degree 18 over $\mathbb{Q}$. $\endgroup$ – carmichael561 Apr 8 '16 at 17:13

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