(d) Z[ /2] is a field.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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.
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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