November 13th, 2017, 12:58 PM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 104 Thanks: 1  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 $\mathbb{Q}$. Thanks 
November 13th, 2017, 08:02 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 609 Thanks: 378 Math Focus: Dynamical systems, analytic function theory, numerics 
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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