Number 2 d and e 1.2 is an integral domain, but not a field.Z, is a field for every prime number p.2.EXERCISE:1.Let be a ring with unity and let A* be the set of all units in A. Show thatthe set is a group.2.Let Z [√2] = { a+b√2 labe Z}. Define addition and multiplication onZ[√2] as follows:(a+b√2)+(c + d√2) (a + c)+(b+d)√2(a+b√2)(c + d√2)= (ac+2bd) + (ad + bc)√2Prove or disprove the following:a)Z [√2] is a ringb)Z [√2] is a commutative ringZ [√2] is a ring with unityZ [√2] is a fieldZ [√2] is an integral domainShow that S = {(a,a) | ae Z) is a subring of Zx Z (Use the definition ofaddition and multiplication in direct products)Determine whether T = {{ (a,- a) | ae Z) is a subring of Zx Z7b)

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Answered: d) ej Z [√2] is a field Z [√2] is an…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 54E

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d) ej Z [√2] is a field Z [√2] is an integral domain