6. Show the ring {a+bv2|z, b € Z} has exactly 2 automorphisms. These are the identity and the map which switches sign on the square root. Show they preserve addition and multiplication. Any automorphism must be linear and must be the identity on the integers. The only question is what happens to the square root. It must go to an element whose square is 2.

6. Show the ring {a+bv2|z, b € Z} has exactly 2 automorphisms. These are the identity and the map which switches sign on the square root. Show they preserve addition and multiplication. Any automorphism must be linear and must be the identity on the integers. The only question is what happens to the square root. It must go to an element whose square is 2.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

6. Show the ring {a+bv/2|z, b E Z} has exactly 2 automorphisms. These
are the identity and the map which switches sign on the square root. Show
they preserve addition and multiplication. Any automorphism must be linear
and must be the identity on the integers. The only question is what happens
to the square root. It must go to an element whose square is 2.

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