Let $p=t^5-6t+3$ be a polynomial. It has two nonreal roots and three real roots. In this video is explained that if $\mathbb{Q}(a_1)/\mathbb{Q}$, where $a_1$ is one of the five roots, is a degree 5 extension since $t^5-6t+3$ is the minimal polynomial and it's irreducible by Eisenstein. Then the video argues that $5\,|\,|\text{Gal}(p)|$ by the 'tower law'.

Now, I know that if $\Sigma$ is the splitting field, then the tower law says that $[\Sigma:\mathbb{Q}]=[\Sigma:\mathbb{Q}(a_1)]\cdot\underbrace{[\mathbb{Q}(a_1):\mathbb{Q}]}_{=\,5}$, but why is it then that $|\text{Gal}(p)|=[\Sigma:\mathbb{Q}]$?

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    $\begingroup$ The Galois group of a polynomial is, by definition, the Galois group of its splitting field. $\endgroup$ – user26857 Oct 28 '15 at 22:33
  • $\begingroup$ and by one of the basic theorems of Galois Theory, the order of the Galois group is the field extension degree of the Galois extension $\Sigma\supset\Bbb Q$. $\endgroup$ – Lubin Oct 29 '15 at 19:27

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