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Cyclic extension

  1. Oct 14, 2008 #1

    PIM

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    1. The problem statement, all variables and given/known data
    Let K be a field, and let K' be an algebraic closure of K. Let sigma be
    an automorphism of K' over K, and let F be the fix field of sigma. Let L/F
    be any finite extension of F.


    2. Relevant equations

    Show that L/F is a finite Galois extension whose
    Galois group Gal(L/F) is cyclic.

    3. The attempt at a solution
     
  2. jcsd
  3. Oct 14, 2008 #2

    PIM

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    I thought about the prime subfiled of F, which is isomorphic to F_p or Q, and tried to prove that L is finite Galois over this prime subfield. (but failed) if I could show this, then it's obvious that L is finite galois over F since F is an intermediate field.

    But for the cylic galois group, I still haven't got any idea.
     
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