For and with we write .

The residue class of modulo is given by the equivialence class .

We denote the set of residue classes mod with .

is a field if and only if is prime.

Proof.

• Let be prime then . thus is a field.
• Assume is a field and with . Then thus has no solution. So , which is contradictory.

We have already established here, that is a cummutative unitary ring. ( TODO)

Euler's totient function is defined by .

If is prime then .

Proof. By definition of the totient function.

Let and with then .

Proof. TODO