November 13th, 2017, 01:58 PM  #1 
Member Joined: Jan 2015 From: usa Posts: 75 Thanks: 0  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, 09:02 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 197 Thanks: 105 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.


Tags 
extension, normal 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
An extension to FLT  magicterry  Number Theory  1  March 31st, 2017 10:00 PM 
why is it a normal field extension?  rayman  Abstract Algebra  3  November 27th, 2013 04:52 AM 
is this field extension normal?  rayman  Abstract Algebra  2  November 26th, 2013 09:37 PM 
normal to the curve/normal to the circle  cheyb93  Calculus  7  October 29th, 2012 02:11 PM 
Product normal implies components normal  cos5000  Real Analysis  3  November 18th, 2009 09:52 PM 