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Homework Help: Degree of Field Extension

  1. May 5, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the splitting field of x8 − 2 over Q, including its degree

    2. Relevant equations
    Degree is multiplicative in tower of fields:
    F [tex]\subset[/tex] K [tex]\subset[/tex] L
    [L:F] = [L:K][K:F]

    Degree of Galois extension is equal to the order of the Galois group
    [K:F] = |Gal(K/F)|

    3. The attempt at a solution
    The splitting field is gotten by adjoining the eighth root of 2 the a primitive eighth root of unity K=Q([tex]\sqrt[8]{2}[/tex], [tex]\zeta[/tex]).

    Q[tex]\subset[/tex]Q([tex]\sqrt[8]{2}[/tex]) [tex]\subset[/tex] K
    [K: Q([tex]\sqrt[8]{2}[/tex])] = 8
    => 8 | [K : Q]

    Q[tex]\subset[/tex]Q([tex]\zeta[/tex]) [tex]\subset[/tex] K
    [K: Q([tex]\sqrt[8]{2}[/tex])] = 7
    => 7 | [K : Q]

    So I believe 56 must divide the order of the field extension. I also know since K is the splitting field of the polynomial it in a Galois extension; so the degree must be the order of the Galois group. However I have tried, but I have had much difficultly getting a handle of the Galois group Gal(K/Q)
  2. jcsd
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