(d) Z[ /2] is a field.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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num 4 help
![4. Let Z[v2] = {a +bV2|a,b e Z}. Define addition and multiplication on Z[ V2] as
follows:
(a+b [v2 ]) + (c+d [ /2 ]) = (a+c) + (b+d)[ V2]
(a+b [ v2 ])(c+d [ /Z1)
(ac+2bd) + (ad+bc) V2
Prove or disprove the following statements:
(a) Z[ /2] is a ring.
(b) Z[ /2] is a commutative ring.
(c) Z[ V2] is a ring with unity.
(d) Z[ /2] is a field.
(e) Z[ /2] is an integral domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc36662-e81e-4805-a9ba-e756ced76ed7%2F4ec3a5f8-07ac-4b2a-bbbb-de8541df38ac%2Fz039v7d_processed.png&w=3840&q=75)
Transcribed Image Text:4. Let Z[v2] = {a +bV2|a,b e Z}. Define addition and multiplication on Z[ V2] as
follows:
(a+b [v2 ]) + (c+d [ /2 ]) = (a+c) + (b+d)[ V2]
(a+b [ v2 ])(c+d [ /Z1)
(ac+2bd) + (ad+bc) V2
Prove or disprove the following statements:
(a) Z[ /2] is a ring.
(b) Z[ /2] is a commutative ring.
(c) Z[ V2] is a ring with unity.
(d) Z[ /2] is a field.
(e) Z[ /2] is an integral domain.
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