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May 30th, 2017, 07:19 AM   #1
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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
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May 31st, 2017, 04:07 AM   #2
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Quote:
Originally Posted by Lalitha183 View Post
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
Please someone help!!!
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June 1st, 2017, 12:36 PM   #3
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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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