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Homework Help: Field theory question

  1. Jan 9, 2007 #1
    1. The problem statement, all variables and given/known data

    Show [itex]Q(\sqrt{p},\sqrt{q}) = Q(\sqrt{p} + \sqrt{q}) [/itex]

    2. Relevant equations

    [itex]p[/itex] and [itex]q[/itex] are two different prime numbers

    3. The attempt at a solution

    I can show [itex]\sqrt{p} + \sqrt{q} \in Q(\sqrt{p},\sqrt{p}) [/itex]

    I have trouble with the other direction though, i.e [itex]\sqrt{p},\sqrt{p} \in Q(\sqrt{p} + \sqrt{q}) [/itex].

    So far I've let [itex] \alpha = \sqrt{p} + \sqrt{q}[/itex]

    and found the powers [itex] \alpha^2 = p + q + 2 \sqrt{p}\sqrt{q}[/itex] and [itex] \alpha^3 = (p + 3q )\sqrt{p} + (3p + q)\sqrt{q}[/itex]

    Not sure what to do now though.
  2. jcsd
  3. Jan 9, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    So you know that
    [itex] \sqrt{p} + \sqrt{q}[/itex]


    [itex] (p + 3q )\sqrt{p} + (3p + q)\sqrt{q}[/itex]

    are in the field. That is all you need. HINT: think simlutaneous linear equations.
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