Exercise 1 Let Zij = (a + bi | a, b E Z}, and define the operation + in Z z] as ordinary addition (as in C). Note that Z = { a + 0i | a E 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, be Z }. Prove that (5) is a normal subgroup of Z[i]. (d) Prove that Z[i]/(5) is an abelian group (with operation +)