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I am convinced that the splitting field over ${\mathbb Q}$ of $f(t)=t^6-10t^3+22$ has degree 36. This is equivalent to prove that $\sqrt[3]{5+\sqrt{3}}$ does not belong to ${\mathbb Q}(\sqrt[3]{5-\sqrt{3}})$ and I am not able to check this. Could anyone help to me?

JOSE MANUEL GAMBOA MUTUBERRIA
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  • The degree of the extension divides the degree of the minimal polynomial IIRC? – Henno Brandsma Aug 13 '21 at 10:49
  • Yes, the Galois group of $f$ has order $36$. Consider the polynomial $x^2-10x+22$, with $x=t^3$. – Dietrich Burde Aug 13 '21 at 10:56
  • I am surely stupid, but I am unable to deduce the result about $t^6-10t^3+22$ after substituting $x=t^3$ in $t^2-10t+22$. Sorry. – JOSE MANUEL GAMBOA MUTUBERRIA Aug 13 '21 at 12:17
  • If both $a,b=\sqrt[3]{5\pm\sqrt {3}}$ belong to the same field extension $\mathbb{Q} (a) $ then $c=(a+b) $ also lies in same field. But $c^3=10+3\sqrt[3]{22}c$ or $c^9-30c^6-294c^3-1000=0$ and then you need to visit this thread :https://math.stackexchange.com/q/4214793/72031 – Paramanand Singh Aug 13 '21 at 13:52
  • Is this related to some question from a textbook? – Paramanand Singh Aug 13 '21 at 14:48
  • I am preparing my lectures on fields extensions and Galois theory and I want to present a non trivial example. Your answer is very helpful, 'many thanks! but I do not know how to prove that $t^9-30t^6-294t^3-1000$ is irreducible in ${\mathbb Z}[t]$. You succeded in the recommended thread but I do not know how to prove the irreducibility of $t^9-30t^6-294t^3-1000$ by hand. – JOSE MANUEL GAMBOA MUTUBERRIA Aug 14 '21 at 10:12
  • I succeed solving my question using Singh trick together with a fantastic explanation of Prof. Jyrki Lahtonen posted Aug. 7. Thanks a lot. ++ for both – JOSE MANUEL GAMBOA MUTUBERRIA Aug 14 '21 at 15:57
  • Glad to know that you were able to solve the problem with the help of that Q&A. If $f(t) =t^9-30t^6-294t^3-1000$ then $g(t) =f(5t)/125\in\mathbb {Z} [t] $ and we can use Murty Criterion on $g(t) $ with $g(9)$ being prime. – Paramanand Singh Aug 16 '21 at 07:35
  • Much more elegant, Paramanand. I should convince my students that g(9) is prime; this is not easy without a machine. Many thanks. – JOSE MANUEL GAMBOA MUTUBERRIA Aug 17 '21 at 09:55

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