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Number fields

  1. Apr 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Find a sequence of extension fields (i.e. tower)
    Q= F[tex]_{0}[/tex][tex]\subseteq[/tex].......[tex]\subseteq[/tex]F[tex]_{n}[/tex].

    where [tex]\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}[/tex] [tex]\in[/tex] F[tex]_{n}[/tex]

    Prove that all the steps are non-trivial. except the last one. btw Q is the set of rational number. and 0 and n on F were meant to be subscripts not superscripts (i dont know how to do that)

    2. Relevant equations



    3. The attempt at a solution

    I'm a bit confused as to what to do in this question? I dont think I understand the question.[tex]\sqrt{}[/tex]
     
    Last edited: Apr 1, 2009
  2. jcsd
  3. Apr 1, 2009 #2

    HallsofIvy

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    The "trivial" step is [itex]F_1= Q(\sqrt{1})[/itex] since [itex]\sqrt{1}= 1[/itex] which already is a rational number. Take [itex]F_2= F_1(\sqrt{2})= Q_(\sqrt{2})[/itex], [itex]F_3= F_2(\sqrt{3})[/itex], etc.
     
    Last edited: Apr 3, 2009
  4. Apr 2, 2009 #3
    ok i think i got it, can anyone please check my answer

    K_0 = 2 which corresponds to F_0
    K_1= 1 + [tex]\sqrt{2}[/tex] for F_1
    K_2= 1 + [tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] for F_2
    K_3= 1 + [tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] + [tex]\sqrt{5}[/tex] for F_3
    K_4= [tex]\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}[/tex] for F_4


    except the last one is supposed to go on forever? can anyone help me in this.
     
  5. Apr 3, 2009 #4

    HallsofIvy

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    Your original question did not "go on forever", it stopped at [itex]\sqrt{5}[/itex].
     
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