Galois Theory Basic Algebra I Jacobson I'm having troubles with this problem.  I saw it in Jacobson's "Basic Algebra I".
I hope you can help me.
-Show that $E=\mathbb{Q}(\sqrt{2}, \sqrt{3}, u)$ where $u^2=(9-5\sqrt{3})(2-\sqrt{2})$ is normal. Determine $\mathrm{Gal}(E/\mathbb{Q})$.
 A: To show that $E/\mathbb Q$ is normal (in fact Galois), it suffices to show that $|\mathrm{Aut}(E/\mathbb Q)|=[E:\mathbb Q]$, so we shall first determine $\mathrm{Aut}(E/\mathbb Q)$ and $[E:\mathbb Q]$. To do so consider the tower of fields $E/\mathbb Q(\sqrt2,\sqrt3)/\mathbb Q(\sqrt2)/\mathbb Q$. We see that each extension in this tower has degree $2$, since they are the splitting fields of $x^2-(9-5\sqrt3)(2-\sqrt2),x^2-3$ and $x^2-2$ respectively (show this), so $[E:\mathbb Q]=2\cdot 2\cdot 2=8$. Each extension has two automorphisms, the identity and the maps
$$\begin{align}
a+bu&\mapsto a-bu &a,b\in\mathbb Q(\sqrt2,\sqrt3)\\
a+b\sqrt3&\mapsto a-b\sqrt3 &a,b\in \mathbb Q(\sqrt2)\\
a+b\sqrt2&\mapsto a-b\sqrt2 &a,b\in\mathbb Q
\end{align}$$
Clearly these automorphisms fix $\mathbb Q$. It remains to check that the last two extend to all of $E$. I leave it to you to check that the maps
$$\begin{align}
a+b\sqrt3&\mapsto a-b\sqrt3 &a,b\in \mathbb Q(\sqrt2,u)\\
a+b\sqrt2&\mapsto a-b\sqrt2 &a,b\in\mathbb Q(\sqrt3,u)
\end{align}$$
are well-defined field automorphisms. This gives you three non-identity automorphisms, which I'll call $\alpha,\beta,\gamma$. It is easy to see that $\mathrm{id}, \alpha,\beta,\gamma,\alpha\beta,\alpha\gamma,\beta\gamma,\alpha\beta\gamma$ are all distinct. Since $|\mathrm{Aut}(E/\mathbb Q)|\leq [E:\mathbb Q]=8$ it follows that these must be all automorphisms of $E/\mathbb Q$, and so $|\mathrm{Aut}(E/\mathbb Q)|=[E:\mathbb Q]$ hence the extension is Galois and
$$\mathrm{Gal}(E/\mathbb Q)=\{\mathrm{id}, \alpha,\beta,\gamma,\alpha\beta,\alpha\gamma,\beta\gamma,\alpha\beta\gamma\}.$$
