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Thread: Prove Q(\sqrt{2}+\sqrt{3})=Q(\sqrt{2},\sqrt{3})

  1. April 11th 2010, 04:26 PM #1
    apple2009
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    Prove Q(\sqrt{2}+\sqrt{3})=Q(\sqrt{2},\sqrt{3})

    Prove Q( +)=Q(,)
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  2. April 11th 2010, 04:52 PM #2
    NonCommAlg
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    Quote Originally Posted by apple2009 View Post
    Prove Q( +)=Q(,)
    let \alpha=\sqrt{2}+\sqrt{3}. then \frac{1}{\alpha}=\sqrt{3}-\sqrt{2} and thus \sqrt{3}=\frac{\alpha^2+1}{2\alpha} \in \mathbb{Q}(\alpha) and \sqrt{2}=\alpha - \sqrt{3} \in \mathbb{Q}(\alpha).
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