The Gaussian integers ℤ[i] = {a + bi : a,b ∈ ℤ} form a unique factorization domain with norm N(a+bi) = a^2 + b^2. Gaussian primes include rational primes p ≡ 3 (mod 4) and factors a±bi of primes p ≡ 1 (mod 4), elegantly explaining two-square representations.
From algebraic integers, you know that ℤ[i] is the ring of integers of ℚ(i) — the "integral" elements of the field of Gaussian rationals. The Gaussian integers look like the ordinary integers ℤ but spread out over the complex plane on a square grid. The unit circle in ℤ[i] contains only four elements — 1, −1, i, −i — corresponding to the four units. Just as in ℤ, units are the invertible elements, and factorization is only unique "up to units."
The key to factorization in ℤ[i] is the norm: N(a + bi) = a² + b² = (a + bi)(a − bi). The crucial property is multiplicativity — N(αβ) = N(α)N(β). This turns the problem of factorization in ℤ[i] into a problem about ordinary integers, because if α factors as βγ then N(α) = N(β)N(γ) in ℤ. In particular, an element with prime norm must be a Gaussian prime (irreducible in ℤ[i]). For example, 2 + i has norm 5, which is a rational prime, so 2 + i is a Gaussian prime.
The fate of rational primes in ℤ[i] falls into three cases that depend on their residue mod 4. A prime p ≡ 3 (mod 4) remains prime — it is still irreducible in ℤ[i]. Geometrically, there is no way to write p = a² + b² as a sum of two squares when p ≡ 3 (mod 4). A prime p ≡ 1 (mod 4) splits: it factors as p = (a + bi)(a − bi) where a² + b² = p — and Fermat's theorem on sums of squares guarantees this factorization exists. The prime p = 2 is special: 2 = −i(1 + i)², so 2 ramifies (its Gaussian factor repeats). This tripartite classification is the prototype for how primes "split, remain inert, or ramify" in general number fields.
The payoff of all this machinery is Fermat's two-square theorem: a positive integer n is expressible as a sum of two squares n = a² + b² if and only if every prime factor of n of the form 4k + 3 appears to an even power. The proof flows directly from unique factorization in ℤ[i]. Writing n as a norm N(a + bi) = a² + b² is exactly asking for n to factor in ℤ[i], and unique factorization tells you exactly when that is possible. A 2,000-year-old theorem about sums of squares becomes a routine consequence of the algebraic structure.
The reason ℤ[i] is particularly tractable is that it is a Euclidean domain: you can perform division with remainder using the norm as the "size" function. Specifically, given α, β ∈ ℤ[i] with β ≠ 0, you can always find γ, ρ ∈ ℤ[i] with α = βγ + ρ and N(ρ) < N(β). This geometric fact — that every complex number lies within distance 1/√2 < 1 of some Gaussian integer — is the reason Euclid's algorithm works in ℤ[i], and it is what ultimately guarantees unique factorization. Not every ring of algebraic integers enjoys this property, making ℤ[i] an especially clean model to master before tackling more complex number fields.