63 N(3)).50. Let R Z Finda. N((0)).c. N(9)).b. N(4)).c. N((6)).nilpotent elements.is coefficients are 0 or 1, and addition and multiplication of coeffi-cients are done modulo 2). Show that Z,[x\/? + x + 1) is a field.4 List the elements of the field given in Exercise 51, and make an ad-dition and multiplication table for the field.55. Show that Z[x]/(x² + x + 1) is not a field.56 Let R be a commutative ring without unity, and let a ER. Describethe smallest ideal I of R that contains a (that is, if J is any ideal thatcontains a, then ICJ).E7 Let R be the ring of continuous functions from R to R. Let A ={fERIf(0) is an even integer}. Show that A is a subring of Rbut not an ideal of R.58. Show that Z[i]/(1 - i) is a field. How many elements does this fiehave?59. If R is a principal ideal domain and I is an ideal of R, prove that cery ideal of R/I is principal (see Exercise 43).60. How many elements are in Z,[i]/(1 + i)?61. Show, by example, that the intersection of two prime idealsnot be a prime ideal.62. Let R denote the ring of real numbers. Determine all ideals of R.What happens if R is replaced by any field F?63. Find the characteristic of Z[i]/{2 + i).64. Show that the characteristic of Z[i]/Ka + bi) divides a² + b²-65. Prove that the set of all polynomials whose coefficients areis a prime ideal in Z[x].%3D00. Let R = Z[V-5] and let I = (a + bV-5I a, b E Z -imal ideal of R.%3D

Name: 63 N(3)).50. Let R Z Finda. N((0)).c. N(9)).b. N(4)).c. N((6)).nilpote
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Description: 63 N(3)).50. Let R Z Finda. N((0)).c. N(9)).b. N(4)).c. N((6)).nilpotent elements.is coefficients are 0 or 1, and addition and multiplication of coeffi-cients are done modulo 2). Show that Z,[x\/? + x + 1) is a field.4 List the elements of the field

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characteristic of Z[i]/(2 + i). nhoracteristic of Z[i]/(c

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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If V=R\power{3}, and W\index{1} is the xy plane and let W\index{2} is yz-plane: W\index{1}={(x,y,0):x,y∈R} W\index{2}={(0,y,z):y,z∈R} then V is not the direct sum of W\index{1} and W\index{2}.
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