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Math Help - galois group

  1. #1
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    galois group

    Determine the Galois group over Q of the splitting field of the question. List all of the subgroups of the Galois group. List all of the subfield of the splitting field of the equation.
    (X^7)-1
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    Determine the Galois group over Q of the splitting field of the question. List all of the subgroups of the Galois group. List all of the subfield of the splitting field of the equation.
    (X^7)-1
    Hint: \text{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q}) \cong (\mathbb{Z}/7\mathbb{Z})^\times \cong \mathbb{Z}/6\mathbb{Z}.

    Find the subgroups of it and correspond them to the intermediate fields.

    For instance, if you choose a generator of the above group as \sigma:\zeta_7 \mapsto \zeta_7^3, then one of the subgroups of it is \{1, \sigma^3\}.

    The reason why I choose the above generator (among them) is because 3 has order 6 in (\mathbb{Z}/7\mathbb{Z})^\times ( Meanwhile, 2 has order 3 in (\mathbb{Z}/7\mathbb{Z})^\times).

    Since \zeta_7 + \sigma^3\zeta_7=\zeta_7 + \zeta_7^{3^3}=\zeta_7 + \zeta_7^{-1}, the corresponding intermediate field is \mathbb{Q}(\zeta_7+\zeta_7^{-1}).

    I'll leave it to you to find the remaining subgroups of <\sigma> and their corresponding intermediate fields.
    Last edited by TheArtofSymmetry; May 19th 2010 at 04:16 PM.
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