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Thread: Elements of Extension Fields

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    Elements of Extension Fields

    Prove that $\displaystyle \sqrt{7}$ is not an element of $\displaystyle \mathbb{Q}(\sqrt{3 + \sqrt{2}}).$
    Last edited by h2osprey; May 19th 2011 at 11:58 AM.
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    Quote Originally Posted by h2osprey View Post
    Prove that $\displaystyle \sqrt{7}$ is not an element of $\displaystyle \mathbb{Q}(\sqrt{3 + \sqrt{2}}).$
    here is one way to solve the problem:
    let$\displaystyle \sqrt{3+\sqrt{2}}=a$ and $\displaystyle \sqrt{3-\sqrt{2}}=b$. if $\displaystyle \sqrt{7} \in \mathbb{Q}(a)$, then $\displaystyle b \in \mathbb{Q}(a)$ because $\displaystyle ab = \sqrt{7}$. that means $\displaystyle \mathbb{Q}(a)$ is the splitting field of $\displaystyle x^4-6x^2+7$. thus $\displaystyle \mathbb{Q}(a)/\mathbb{Q}$ is Galois and hence $\displaystyle |\text{Gal}(\mathbb{Q}(a)/\mathbb{Q})|=[\mathbb{Q}(a):\mathbb{Q}]=4.$ this is a contradiction because the galois group of $\displaystyle x^4-6x^2+7$ is the dihedral group of order $\displaystyle 8$.
    Last edited by NonCommAlg; May 19th 2011 at 05:56 PM.
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