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Prove that the extension is Normal

  1. Feb 13, 2013 #1
    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?
  2. jcsd
  3. Feb 15, 2013 #2
    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 [itex] u^8-2496u^4+2304[/itex], though you should check my work. Of course, you then need to show that [itex] \sqrt 2, \sqrt 3, u [/itex] generate the roots, and that no subextension will work.
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