The nonzero elements of $\displaystyle Z_{3}[i]$ form an Abelian group of order 8 under multiplication. Is it isomorphic to $\displaystyle Z_{8}, Z_{4} \oplus Z_{2}, Z_{2} \oplus Z_{2} \oplus Z_{2}$?
The nonzero elements of $\displaystyle Z_{3}[i]$ form an Abelian group of order 8 under multiplication. Is it isomorphic to $\displaystyle Z_{8}, Z_{4} \oplus Z_{2}, Z_{2} \oplus Z_{2} \oplus Z_{2}$?
It is $\displaystyle \mathbb{Z}_8$.*
*)A classic result is that the multiplicative group of a finite field must be cyclic. That is the only cyclic group on that list.