Exercise 1 Let Z[i] = {a+bi | a,b€ Z}, and define the operation + in Z[i] as ordinary addition (as in C).

Note that Z= { a+0i | a € Z } is naturally a subset of Z[i].

(a) Prove that (Z, +) is a normal subgroup of (Z[i], +).

(b) Prove that the quotient group Z[i]/Z (with operation +) is an abelian group.

(c) Let (5) denote the set (5) = { 5a +5bi | a,b € Z}. Prove that (5) is a normal subgroup of Z[i].

(d) Prove that Z[i]/(5) is an abelian group (with operation +).