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.