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Smallest subfields containing Z and Z[sqrt2]

  1. Feb 10, 2010 #1
    Smallest subfields containing Z and Z[sqrt2]

    1. The problem statement, all variables and given/known data
    What are the smallest subfields of R containing Z and Z[[tex]\sqrt{2}[/tex]]?


    2. Relevant equations
    Z= {a+ib|a,b[tex]\in Z[/tex]}
    Z[[tex]\sqrt{2}[/tex]]={a+b[tex]\sqrt{2}[/tex]|a,b[tex]\in Z[/tex]}

    3. The attempt at a solution
    Z[tex]\subset[/tex]Q and Z[[tex]\sqrt{2}[/tex]][tex]\subset[/tex]Q[[tex]\sqrt{2}[/tex]]
    so to get the subfield that contains both of them, would it be Q[i,[tex]\sqrt{2}[/tex]]?
     
  2. jcsd
  3. Feb 10, 2010 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: Smallest subfields containing Z and Z[sqrt2]

    Yes.
     
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