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Galois correspondence

  1. Jan 10, 2007 #1
    1. The problem statement, all variables and given/known data

    i) Find the order and structure of the Galois Group [itex]K:Q[/itex] where
    [itex] K = Q(\alpha) [/itex] and

    [itex] \alpha = \sqrt{2 + \sqrt{2}}[/itex].

    ii)Then for each subgroup of [itex]Gal (K:Q)[/itex], find the corresponding subfield through the Galois correspondence.

    2. Relevant equations

    I get the minimal polynomial to be [itex] f(x): x^4 -4x^2 + 2[/itex] and the four roots of [itex] f(x)[/itex] are [itex] \sqrt{2 + \sqrt{2}} , -\sqrt{2 + \sqrt{2}}, \sqrt{2 - \sqrt{2}}, -\sqrt{2 - \sqrt{2}} [/itex]

    I'm told [itex]K:Q[/itex] is normal

    3. The attempt at a solution

    I get the order of [itex]Gal (K:Q)[/itex] to be 4 with

    [itex] Gal (K:Q) = \{ Id, \sigma, \tau, \sigma \tau \} [/itex] where

    [itex] \sigma(\sqrt{2 + \sqrt{2}} ) \rightarrow -\sqrt{2 + \sqrt{2}} [/itex]

    [itex] \tau(\sqrt{2 + \sqrt{2}} ) \rightarrow \sqrt{2 - \sqrt{2}} [/itex]

    and

    [itex]\sigma \tau (\sqrt{2 + \sqrt{2}} ) \rightarrow -\sqrt{2 - \sqrt{2}} [/itex]

    I get stuck with the 2nd part though. I get the proper subgroups [itex] \{Id, \sigma \} \{Id, \tau\} \{ Id, \sigma \tau \} [/itex] but I don't see how to find the corresponding subfields.

    Through the Galois correspondence, there's a bijective map between the subfields of K and the subgroups of [itex] Gal(K:Q)[/itex]. So there'd have to be 3 different subfields. I know these fields have to be generated by the fixed points for each map.

    I'd appreciate any help on what the subfields are.
     
    Last edited: Jan 10, 2007
  2. jcsd
  3. Jan 10, 2007 #2

    Hurkyl

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    Well, you know how to find fixed points of any linear transformation, don't you? They're eigenvectors with eigenvalue 1.



    Alternatively... finding 1 fixed point is easy. You know [itex]\alpha[/itex] isn't fixed by, say, [itex]\sigma[/itex]. But can you arrange for

    [itex]\alpha[/itex] + stuff = [itex]\sigma(\alpha)[/itex] + other stuff?

    (Just to be clear, you should not be solving any equations with this approach)

    I don't know if this automatically gives you a generator, though. You'll have to prove that this is a generator through some other means.
     
    Last edited: Jan 10, 2007
  4. Jan 10, 2007 #3
    Sorry, I don't understand this.

    Is finding the fixed point of [itex] \sigma (\alpha)[/itex] something to do with the sign in between the [itex]2 [/itex] and [itex] \sqrt{2}[/itex] not changing?
     
  5. Jan 10, 2007 #4

    Hurkyl

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    (I will use s for sigma, and a for alpha) You want to find "something" such that

    something = s(something).

    Well, what might appear in "something"? a might. So, we try it out:

    a + other stuff = s(a + other stuff) = s(a) + s(other stuff).

    Doesn't that suggest what might appear in "other stuff"?
     
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