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November 13th, 2017, 12:58 PM   #1
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Normal extension

Please help me to answer the following problem:

Let $\alpha=\sqrt{2+\sqrt{3}}$.

Let $L=\mathbb{Q}(\alpha)$. Show that $L$ is a normal extension of $\mathbb{Q}$.

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November 13th, 2017, 08:02 PM   #2
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The Galois group for this extension is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z_2}$ (prove this). An extension is normal if and only if the subgroup of automorphisms which fix each intermediate field is a normal subgroup of the Galois group. When this group is abelian, every subgroup is normal and thus every extension is Galois.
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