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Degree of a field extension

  1. Apr 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Let [itex]ζ=e^((2*\pi*i)/7), E=Q(ζ), \xi=ζ + ζ^6[/itex].

    Show that [itex][Q(ζ):Q(\xi)]=2[/itex].
    Find the generator of the galois group [itex]Gal(Q(ζ):Q(\xi))[/itex].
    What is the minimal polynomial of [itex]\xi[/itex].

    2. Relevant equations



    3. The attempt at a solution
    I know that [itex][Q(ζ):Q]=6[/itex] and that [itex]Gal(Q(ζ):Q)[/itex] is the cyclic group of order six. So I need to show that [itex][Q(\xi):Q]=3[/itex], since [itex][Q(ζ):Q]=[Q(ζ):Q(\xi)][Q(\xi):Q]=6=2*3[/itex].
    Using [itex]ζ^6+ζ^5+ζ^4+ζ^3+ζ^2+ζ+1=0[/itex] and dividing by [itex]ζ^3[/itex] we get [itex]ζ^3+ζ^2+ζ+1+ζ^-1+ζ^-2+ζ^-3=0=(ζ+ζ^-1)^3+(ζ+ζ^-1)^2-2(ζ+ζ^-1)-1=\xi^3+\xi^2-2\xi-1[/itex]. Not sure how to proceed from here. Any help would be greatly appreciated :).


    edit: Whats wrong with my latex?
     
    Last edited: Apr 27, 2013
  2. jcsd
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