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Prove equality of number fields

  1. Dec 1, 2008 #1
    Hello everyone,

    I need to prove that Q[i + sqrt(2)] = Q[squrt(2)]

    where Q = rationals

    Any help would be appreciated.

    Thanks
     
  2. jcsd
  3. Dec 1, 2008 #2

    Dick

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    Prove i+sqrt(2) is in Q[sqrt(2)]. That's easy. Then prove i and sqrt(2) are in Q[i+sqrt(2)]. That's a little harder, but not much.
     
  4. Dec 1, 2008 #3

    Hurkyl

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    Degree counting could work too.
     
  5. Dec 1, 2008 #4
    thank you for taking the time to reply.

    With degree counting, would that be counting the bases? Unforunately my professor did not explain this well and I am having a difficult time finding information regarding this subject.
     
  6. Dec 1, 2008 #5

    Hurkyl

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    I'm not sure what you mean by "counting the bases". I'm referring to looking at the degrees of various field extensions.
     
  7. Dec 1, 2008 #6
    a couple of things...

    How exactly do I go about find the various field extensions and once I do, how does this help me prove equality?
     
  8. Dec 1, 2008 #7

    Dick

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    I'm not really sure what Hurkyl is up to, but just try the direct approach. Show Q[i+sqrt(2)] is a subset of Q[sqrt(2)] and conversely.
     
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