How do I check if b) is an ideal? I think it is principal and prime so it must be an ideal but I don't think it's maximal since it's reducible?Thank you very much for the help!! Question 3 In each of the following cases, check whether the given subset is (1) an ideal, (2)a maximal ideal, (3) a principal ideal, and (4) a prime ideal. Justify your answer.(a) I = {0,2} in Z4.(b) U = {q(x) ≤ Z3[x] : x² +x+ 1 is a factor of q(x)} in Z3 [x].E(c) J = {a+i·b: 17a – 190a²b² > 0} in Q(i) = {a+i⋅ b : a,b ≤ Q}, where i E C is theimaginary unit (i² = −1).

As asked we shall solve only sub-part b

Answered: (b) U = {q(x) € Z3 [x] : x²+x+1 is a…

Math

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(b) U = {q(x) € Z3 [x] : x²+x+1 is a factor of q(x)} in Z3 [x].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.4: Maximal Ideals (optional)

Problem 16E

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How do I check if b) is an ideal? I think it is principal and prime so it must be an ideal but I don't think it's maximal since it's reducible?

Thank you very much for the help!!

Question 3 In each of the following cases, check whether the given subset is (1) an ideal, (2)
a maximal ideal, (3) a principal ideal, and (4) a prime ideal. Justify your answer.
(a) I = {0,2} in Z4.
(b) U = {q(x) ≤ Z3[x] : x² +x+ 1 is a factor of q(x)} in Z3 [x].
E
(c) J = {a+i·b: 17a – 190a²b² > 0} in Q(i) = {a+i⋅ b : a,b ≤ Q}, where i E C is the
imaginary unit (i² = −1).

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