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Math Help - Quadratic field question

  1. #1
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    Quadratic field question

    Consider Q[sqrt(2)]. Does every element of Q[sqrt(2)] have a square root in Q[sqrt(2)]? Prove if true; and give a counterexample if false.

    I'm thinking it is true, but I can't prove it. Please help!
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  2. #2
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    Quote Originally Posted by machack View Post
    Consider Q[sqrt(2)]. Does every element of Q[sqrt(2)] have a square root in Q[sqrt(2)]? Prove if true; and give a counterexample if false.

    I'm thinking it is true, but I can't prove it. Please help!

    Suppose \sqrt{\sqrt{2}}\in\mathbb{Q}(\sqrt{2})\Longrightar  row (a+b\sqrt{2})^2=\sqrt{2} , for some a,b\in\mathbb{Q} , but then:

    a^2+2b^2+2\sqrt{2}ab=\sqrt{2}\Longrightarrow make some order here and show this would imply \sqrt{2}\in\mathbb{Q} or some other harsh contradiction .

    Tonio
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