We begin by defining how to perform basic arithmetic modulo $n$, where $n$ is a positive integer. Addition, subtraction, and multiplication follow naturally from their integer counterparts, but we have complications with division.

We will need this algorithm to fix our problems with division. It was originally designed to find the greatest common divisor of two numbers.

Once armed with Euclid’s algorithm, we can easily compute divisions modulo $n$.

We find we only need to study ${\mathbb{Z}}_{p^k}$ where $p$ is a prime, because once we have a result about the prime powers, we can use the Chinese Remainder Theorem to generalize for all $n$.

While studying division, we encounter the problem of inversion. Units are numbers with inverses.

The behaviour of units when they are exponentiated is difficult to study. Modern cryptography exploits this.

If we start with a unit and keep multiplying it by itself, we wind up with 1 eventually. The order of a unit is the number of steps this takes.

We discuss a fast way of telling if a given number is prime that works with high probability.

Sometimes powering up a unit will generate all the other units.

We focus only on multiplication and see if we can still say anything interesting.

Elements of ${\mathbb{Z}}_n$ that are perfect squares are called quadratic residues.