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About a root in the topic of automorphism and fixed field. HELP

  1. Jun 1, 2012 #1
    About a root in the topic of automorphism and fixed field. HELP!!

    1. The problem statement, all variables and given/known data
    Let [itex] m [/itex] be a positive integer.

    Let [itex] \xi = \exp (2pi/m) [/itex]. Then [itex] \xi [/itex] is a primitive m-th root of unity. (I.e., [itex] \xi [/itex] is a solution of

    [itex] \Phi_{m}(X):=(X^m - 1)/(X-1) [/itex].)

    If [itex] \phi \in \mbox{G}(\mathbb{Q}(\xi)/\mathbb{Q}) [/itex], i.e., [itex] \phi [/itex] is an automorphism of [itex]\mathbb{Q}(\xi)[/itex] fixing the elements in [itex]\mathbb{Q}[/itex], then [itex] \phi (\xi) = \xi^i [/itex] for some [itex] i [/itex] with [itex] \gcd (i,m) = 1[/itex]


    2. Relevant equations

    [itex] \mathbb{Q}(\xi) = \mathbb{Q}(\xi^i)[/itex] for any [itex] i [/itex] with [itex] \gcd(i,m)=1 [/itex] but not with [itex] i [/itex] such that [itex] \gcd(i,m) \neq 1 [/itex]



    3. The attempt at a solution

    I know the above relevant equation holds. And also I know that [itex] \phi(\xi) = \xi^i [/itex] for some integer [itex] i [/itex] but I cannot prove [itex] \gcd (i,m) = 1 [/itex]. I've tried to use Isomorphism Extension Theorem, but not really works.
     
    Last edited: Jun 1, 2012
  2. jcsd
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