Prove that the extension is Normal

[Oster]

Prove that the field Q(√2,√3,u) where u^2=(9-5√3)(2√2) is normal over Q.
I'm supposed to show that this field is the splitting field of some polynomial over Q. u is clearly algebraic over Q. Do i just take the higher powers of u and try to find the minimal polynomial over Q or is there a smarter way to do this?

 

[Kreizhn]

Your first idea seems reasonable. There are various definitions of normal extensions, but the one most amenable to actually showing normality is showing that the extension is a splitting field.

I found that the minimal polynomial of u is $u^8-2496u^4+2304$, though you should check my work. Of course, you then need to show that $\sqrt 2, \sqrt 3, u$ generate the roots, and that no subextension will work.