Let Z3i denote the set of nonzero elements of Z3i Compute th

Let Z_3[i]* denote the set of non-zero elements of Z_3[i]. Compute the powers of 1 + i and verify that it generates the set Z_3[i]*. (Thus. Z_3[i]* is a cyclic group under multiplication, generated by 1 + i.)

Solution

Z₃(i) is a cyclic group of order 3.

(1+i)⁰ = 1

(1+i)¹ = 1+ i

(1+i)³ = 1- i

Therefore,

Z₃(i) = { 1, 1+i, 1-i }

Thus, Z³(i) is a cyclic group generated by 1+i.