abstract algebra - how to compute its Galois group

suppose $E=Q(\sqrt{2},\sqrt3,u)$ ,where $u^2=(9-5\sqrt{3})(2-\sqrt{2}）$ The question is to prove it is Galois extension,and compute its Galois group.

Notice that the characteristic of $Q$ is infinity, so very irreducible polynomial is separable. Then just to prove $E/Q$ is the splitting field of the minimal polynomial of $ \sqrt{2},\sqrt3,u$.
My problem is compute the minimal polynomial of $u$,is there any easy way to compute its minimal polynomial?I think only know its minimal polynamial,then I can compute its roots ,which are possible image that $Q-$homophorsims send $u$ to