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 $|\text{Gal}(p)|=[\Sigma:\mathbb{Q}]$?