What if we enlarge the integers to include i = \sqrt{-1}? Gauss did, and found that the resulting numbers a + bi — the Gaussian integers — have their own primes, their own unique factorisation, and they explain whole swathes of ordinary number theory at a glance.

A new number system

The Gaussian integers are \mathbb{Z}[i] = \{\,a + bi : a, b \in \mathbb{Z}\,\} — the lattice of points with integer coordinates in the complex plane. They add and multiply like complex numbers and stay inside the lattice. The key tool is the norm:

which is multiplicative, N(zw) = N(z)N(w) — this is Brahmagupta's identity from the two-squares page. The norm turns questions about Gaussian integers into questions about ordinary integers.

Gaussian primes and unique factorisation

Some ordinary primes stop being prime here. Because 5 = (2 + i)(2 - i), the number 5 splits — it is no longer a Gaussian prime. The pattern mirrors the two-squares theorem exactly:

Crucially, \mathbb{Z}[i] still enjoys unique factorisation (it is a Euclidean domain — Euclid's algorithm works using the norm). That is what makes the two-squares theorem fall out instantly: p is a sum of two squares iff it splits, iff p \equiv 1 \pmod 4.

The bigger idea

Gauss's move — adjoin a new algebraic number and study its arithmetic — is the template for all of algebraic number theory. Sometimes unique factorisation survives the jump (as here); sometimes it spectacularly fails — and rescuing it is what the final stage of this course is about.