Gaussian integers, factorization failure, and norms
In Z, every integer factors uniquely into primes. We take this for granted, but it is a special property called unique factorization, and many rings lack it. A ring where every element factors uniquely into irreducibles is called a unique factorization domain (UFD).
In this lesson, you will explore the Gaussian integers Z[i] (a UFD with a beautiful geometry), see factorization fail dramatically in Z[√-5], and learn how the norm function is the key tool for understanding divisibility.
The Gaussian integers Z[i] = {a + bi : a, b ∈ Z}form a lattice in the complex plane. They are a Euclidean domain (hence a UFD), and their primes have a geometric characterization. Ordinary primes behave in three ways: some split into two Gaussian primes, some ramify, and some stay prime.
Click any lattice point on Z[i] to see its norm and whether it is a Gaussian prime.
Key insight: A prime p in Z splits in Z[i] if p ≡ 1 (mod 4), ramifies if p = 2, and stays prime (inert) if p ≡ 3 (mod 4). This trichotomy connects number theory to the geometry of the complex plane.
In Z[√-5], the number 6 has two genuinely different factorizations: 6 = 2 × 3 = (1+√-5)(1-√-5). All four factors are irreducible, but 2 is not prime: it divides the product (1+√-5)(1-√-5) without dividing either factor. This is the hallmark of non-unique factorization.
Key insight: In a UFD, every irreducible element is prime. In Z[√-5], irreducible elements are not necessarily prime, and this gap is exactly why unique factorization fails. The fix is to use ideals instead -- ideal factorization is always unique in Dedekind domains.
The norm N(a+bi) = a² + b² is a multiplicative function from Z[i] to Z: N(αβ) = N(α)N(β). This makes it a powerful tool for studying divisibility. If N(α) is prime in Z, then α must be irreducible (and prime) in Z[i]. The norm reduces questions about Gaussian integers to questions about ordinary integers.
Adjust sliders to explore Gaussian integers 3+2i. The dashed circle has radius √N = √13. The norm is multiplicative: N(\u03B1\u03B2) = N(\u03B1)\u00B7N(\u03B2).
Key insight: Multiplicative norms are the primary tool for proving irreducibility in algebraic number rings. If no element has a given norm value, then no factorization of that size exists.