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Fixed fields and Galois subgroups.

  1. May 9, 2013 #1
    1. The problem statement, all variables and given/known data

    I need to find all the intermediate fields between [itex]Q(\zeta_7)[/itex] and Q.

    2. Relevant equations



    3. The attempt at a solution

    For the group [itex]Q(\zeta_7)[/itex]. I found all the permutations that are possible for [itex]\zeta_7[/itex].

    Here is what I have:

    Since [itex]Aut(<\zeta_7>) = Z^{\times}_7 = \{1, 2, 3, 4, 5, 7\}[/itex], we know that we can have 6 automorphisms.

    Let [tex]\zeta = \zeta_7[/tex]

    [tex]\zeta \rightarrow \zeta[/tex]

    [tex]\zeta \rightarrow \zeta^2[/tex]

    [tex]\zeta \rightarrow \zeta^3[/tex]

    [tex]\zeta \rightarrow \zeta^4[/tex]

    [tex]\zeta \rightarrow \zeta^5[/tex]

    [tex]\zeta \rightarrow \zeta^6[/tex]

    I computed the orders of these automorphisms (since I want to know the subgroup orders in the Galois group in order to determine the fixed fields).

    So...

    [tex] \zeta \rightarrow \zeta [/tex] has order 1

    [tex] \zeta \rightarrow \zeta^2 \rightarrow \zeta^4 \rightarrow \zeta [/tex] Order=3

    [tex] \zeta \rightarrow \zeta^3 \rightarrow \zeta^2 \rightarrow \zeta^^6 \rightarrow^4 \rightarrow \zeta^5 \rightarrow \zeta [/tex] Order = 6


    [tex] \zeta \rightarrow \zeta^4 \rightarrow \zeta^2 \rightarrow \zeta[/tex] Order = 3

    [tex] \zeta \rightarrow \zeta^5 \rightarrow \zeta^4 \rightarrow \zeta^6 \rightarrow \zeta^2 \rightarrow \zeta^3 \rightarrow \zeta [/tex] Order = 6

    [tex]\zeta \rightarrow \zeta^6 \rightarrow \zeta[/tex] Order = 2

    So now I need to find the fixed subgroups.

    I know that [itex]< \zeta>[/itex] will correspond to [itex]Q(\zeta)[/itex] since it fixes everything.

    I know that the other fields I need to look at are [itex]Q(\zeta^2)[/itex], [itex]Q(\zeta^3)[/itex], [itex]Q(\zeta^4)[/itex], [itex]Q(\zeta^5)[/itex], and [itex]Q(\zeta^6)[/itex].

    There are two reasons why I was stuck:

    1) Since, for an intermediate field K and Galois subgorup H of G, [itex] [ K : Q] = [G : H] [/itex], finding [K:Q] would help reduce the possibilities, right? But I'm kind of confused about how we would be able to find that.

    2) Even if we did find [K:Q] for each K, that would only reduce the possibilites and not tell us exactly which subgorup corresponds to which intermediate field, right? So we have to check which subgroup fixes [itex]\zeta^k[/itex] for some k between 1 and 6, right? But I don't really get anywhere with this method. I'm probably doing something wrong, but I'm not sure what. For example, for [itex]\zeta^3[/itex], I tried to see which automorphism would give me [itex]\zeta^3[/itex] back but couldn't really find any that would. So I'm probably missing something, but I'm not sure why...

    Thank you in advance
     
  2. jcsd
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