May 30th, 2017, 08:19 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  Automorphism
The number of Automorphisms of $\displaystyle Z[\sqrt{2}] $ is A) 1 B) 2 C) 4 D) Infinite I guess its option B. Please check. Thank you 
May 31st, 2017, 05:07 AM  #2 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  
June 1st, 2017, 01:36 PM  #3 
Member Joined: May 2017 From: Russia Posts: 33 Thanks: 4 
Yes, Option B is true. Automorphism $\displaystyle \sigma$ maps 1 to 1. It implies the automorphism fixes $\displaystyle \mathbb{Z}.$ And 0 is mapped to 0. And we have $\displaystyle \sigma\left(\left(\sqrt{2}\right)^22\right)=\left(\sigma\left(\sqrt{2}\right)\right)^ 22=0.$ There are two variants only: either $\displaystyle \sigma(\sqrt{2})=\sqrt{2}$ or $\displaystyle \sigma(\sqrt{2})=\sqrt{2}.$ 

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