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Math Help - Elements of Extension Fields

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    Elements of Extension Fields

    Prove that \sqrt{7} is not an element of \mathbb{Q}(\sqrt{3 + \sqrt{2}}).
    Last edited by h2osprey; May 19th 2011 at 12:58 PM.
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    Quote Originally Posted by h2osprey View Post
    Prove that \sqrt{7} is not an element of \mathbb{Q}(\sqrt{3 + \sqrt{2}}).
    here is one way to solve the problem:
    let  \sqrt{3+\sqrt{2}}=a and \sqrt{3-\sqrt{2}}=b. if \sqrt{7} \in \mathbb{Q}(a), then b \in \mathbb{Q}(a) because ab = \sqrt{7}. that means  \mathbb{Q}(a) is the splitting field of x^4-6x^2+7. thus \mathbb{Q}(a)/\mathbb{Q} is Galois and hence |\text{Gal}(\mathbb{Q}(a)/\mathbb{Q})|=[\mathbb{Q}(a):\mathbb{Q}]=4. this is a contradiction because the galois group of x^4-6x^2+7 is the dihedral group of order 8.
    Last edited by NonCommAlg; May 19th 2011 at 06:56 PM.
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