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]. (c) J = {a+i.b: 17a - 190a²b²2 > 0} in Q(i) = {a+i·b: a,b € Q}, where i EC is the imaginary unit (i² = −1).
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]. (c) J = {a+i.b: 17a - 190a²b²2 > 0} in Q(i) = {a+i·b: a,b € Q}, where i EC is the imaginary unit (i² = −1).
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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![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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F138da6ab-7efa-4b8c-b154-28405afcc8c9%2F6d3ded50-5dc0-4b8b-8853-296983a0688a%2Fwch3tvr_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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