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Understanding fixed fields (Galois Theory)

  1. Dec 2, 2016 #1
    1. The problem statement, all variables and given/known data
    Let's look at Q(a,c):Q where a is the third real root of 2 and c is a primitive cube root of unity. Then this extension is Galois and it's Galois group is isomorphic to ##D_3=S_3##. The proper subgroups of the Galois group are thus <(12)>,<(23)>,<(23)>,<(123)>. Let (12) switch the roots a and ab, (13) switch the roots a and ab^2, (23) switch the roots b and b^2 and (123) be the 3 cycle that will move all roots.

    I'm having trouble understanding the intermediate fields that these groups correspond to.

    2. Relevant equations

    3. The attempt at a solution
    So <(12)> will switch the roots a and ab, thus a+ab must be an element of the fixed field, as must a*ab. is the entire fixed field of the subgroup <(12)> Q(a+ab) and a*ab is an element of Q(a+ab) somehow? But then again, won't b^2 be fixed by this transposition of (12)? I'm picture the complex plane where a is on the positive x axis and b and b^2 are complex numbers with positive and negative imaginary components respectively on the left side of the y axis.

    And then what would the subgroup <(23)> fix? This is the transpotion that corresponds to complex conjugation, so surely b+b^2 would be fixed, but b+b^2=-1 so this is part of Q. This transposition must also fix a, so is the fixed field <(23)> simply Q(a)?

    And then I'm having the same problem with <(13)> as I am with <(12)>. I mean surely <(13)> must fix a+ab^2 as it sends them to eachother, but would it fix ab also? My intuition tells me that the fixed field of <(13)> is just Q(a+ab^2) but I'm not exactly sure why.

    Then I think <(123)> would fix Q(a+ab+ab^2).

    I'd appreciate any feedback and clarification on what's going on here. I'd prefer to understand this in terms of permutations in ##S_3##, but I wouldn't mind understanding it in terms of elements of ##D_3## also.
  2. jcsd
  3. Dec 7, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
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