As $u$ satisfies the polynomial $X^2-(9-5\sqrt3)(2-\sqrt2)$ over $\Bbb{Q}(\sqrt2,\sqrt3)$ and it is irreducible over $\Bbb{Q}(\sqrt2,\sqrt3)$. We have $[\Bbb{Q}(\sqrt2,\sqrt3,u):\Bbb{Q}(\sqrt2,\sqrt3)]=2$. Hence $\Bbb{Q}(\sqrt2,\sqrt3,u)$ is normal over $\Bbb{Q}(\sqrt2,\sqrt3)$. Again, $\Bbb{Q}(\sqrt2,\sqrt3)$ being splitting field of $(X^2-2)(X^2-3)$ over $\Bbb{Q}$, $\Bbb{Q}(\sqrt2,\sqrt3)$ is normal over $\Bbb{Q}$. But it doesn't imply that $\Bbb{Q}(\sqrt2,\sqrt3,u)$ is normal over $\Bbb{Q}$.
I am trying to show $\Bbb{Q}(\sqrt2,\sqrt3,u)$ is splitting field of some polynomial over $\Bbb{Q}$, but not getting as minimal polynomial of $u$ over $\Bbb{Q}$ is coming to be complicated.
Can anyone help me in this regard? Thanks for help in advance.