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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Answered: Recall that the Gaussian integers Z[i]…

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.