Question

Given a natural number q ≥ 1, define a relation ∼ on the set Z by x ∼ y
if x - y is divisible by q.
(i) Show that ∼ is an equivalence relation.
We will denote the set of equivalence classes defined by ∼ with Z=qZ. Also
let x mod q denote the equivalence class to which an integer x belongs.
(ii) Check that the operations

(x (x

mod q) + (y mod q) · (y

mod q) = (x + y) mod q) = (x · y)

mod q; mod q;

are well-defined on Z=qZ.
(iii) With the operations as defined above, show that Z=qZ is not a field
if q is not prime.

Answer 1