Recall that the Gaussian integers Z[i] is the following subring of C: Z[i] = {a+bi a,b ≤ Z}, where as always i denotes the imaginary unit. Use the fact that (1+i)(1−i) = 2 to show that the ideal of Z[i] generated by 2 is not a prime ideal.
Recall that the Gaussian integers Z[i] is the following subring of C: Z[i] = {a+bi a,b ≤ Z}, where as always i denotes the imaginary unit. Use the fact that (1+i)(1−i) = 2 to show that the ideal of Z[i] generated by 2 is not a prime ideal.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 29E: 29. Let be the set of Gaussian integers . Let .
a. Prove or disprove that is a substring of .
...
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![Recall that the Gaussian integers Z[i] is the following subring of C:
Z[i] = {a+bi a,b ≤ Z},
where as always i denotes the imaginary unit. Use the fact that (1+i)(1−i) =
2 to show that the ideal of Z[i] generated by 2 is not a prime ideal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d65aaa5-5279-4ebf-b541-f161ae051282%2Fd73e09ac-8923-49ec-85b0-e553ae7a38b4%2Foocw2ap_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Recall that the Gaussian integers Z[i] is the following subring of C:
Z[i] = {a+bi a,b ≤ Z},
where as always i denotes the imaginary unit. Use the fact that (1+i)(1−i) =
2 to show that the ideal of Z[i] generated by 2 is not a prime ideal.
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