Gaussian integers, factorization failure, and norms
In the integers, every number breaks into primes in exactly one way. Step into other number systems and watch that guarantee hold, and then break.
Click any lattice point to see its norm and whether it is a Gaussian prime. Use the presets to watch 5 split, 2 ramify, and 3 stay prime.
Click any lattice point on Z[i] to see its norm and whether it is a Gaussian prime.
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).
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.
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.
Here 6 factors into irreducibles in two different ways. Click each factor to see why it cannot be broken down further.
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.
Slide a and b to choose a Gaussian integer and watch its norm. Check that norms multiply: N(αβ) = N(α)N(β).
Adjust sliders to explore Gaussian integers 3+2i. The dashed circle has radius √N = √13. The norm is multiplicative: N(αβ) = N(α)·N(β).
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.
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.
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