Gaussian integers

Description

The ring of Gaussian integers is $$\mathbb{Z}[i] = \{a + b i : a, b \in \mathbb{Z}\},$$ where $i \in \mathbb{C}$ satisfies $i^2 = -1$. It is a subring of $\mathbb{C}$, with $(a + b i)(c + d i) = (a c - b d) + (a d + b c) i$. As a subring of a field, it is an integral domain.

Remarks

The norm $N(a + b i) = a^2 + b^2$ is multiplicative. The [[units of $\mathbb{Z}[i]$|06RH4XAGRY]] are $1, -1, i, -i$. The ring $\mathbb{Z}[i]$ is a Euclidean domain, so it has unique factorization. A prime number $p$ is a sum of two squares if and only if $p$ is not prime in $\mathbb{Z}[i]$. This holds exactly for $p = 2$ and the primes $p \equiv 1 \pmod 4$.

Explanations

L1 human

The Gaussian integers are the points with integer coordinates in the complex plane. You add and multiply them as complex numbers. They form a ring with a theory of primes and factorization, like the integers.