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 May 30th, 2017, 07:19 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, 04:07 AM   #2
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 Originally Posted by Lalitha183 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

 June 1st, 2017, 12:36 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 $\displaystyle \sigma\left(\left(\sqrt{2}\right)^2-2\right)=\left(\sigma\left(\sqrt{2}\right)\right)^ 2-2=0.$ There are two variants only: either $\displaystyle \sigma(\sqrt{2})=\sqrt{2}$ or $\displaystyle \sigma(\sqrt{2})=-\sqrt{2}.$

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