Exercise 2 Over the course of the parts of this exercise you will show that multiplication of cosets in Z[i]/Z is not well-defined. (a) Let a, a, b, VEZ. Prove that a +i and a' + i represent the same coset in Z[i]/Z; that is, prove that (a + i) + Z= (a' + i) + Z (b) Prove that band & represent the same coset in Z[i]/Z; that is, prove that b+z=b'+Z (c) Determine the conditions under which (a +i)b and (a' + i)b' represent the same coset in Z[i]/Z. That is, determine what additionally must be assumed in order to ensure that (a + i)b+Z= (a' + i)b' +Z. Be sure to prove your condition is the correct one. (d) Find specific values of a, a, b, b' € Z for which (a+i)b+Z‡ (a¹ + i)b +Z.

Solved: Exercise 2 Over the course of the parts of this exercise you will show

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exercise 2 over the course of the parts of this exercise you will show

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Exercise 2 Over the course of the parts of this exercise you will show that multiplication of cosets in Z[i]/Z is not well-defined. (a) Let a, a, b, VEZ. Prove that a +i and a' + i represent the same coset in Z[i]/Z; that is, prove that (a + i) + Z= (a' + i) + Z (b) Prove that band & represent the same coset in Z[i]/Z;