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The norm N: K → ℚ of a number field K is a multiplicative function sending each element to the product of its conjugates. The norm is essential for factorization properties, ideal theory, and solving Diophantine equations in algebraic number fields.
Start with the simplest nontrivial case: the Gaussian integers ℤ[i], where elements are a+bi with a,b ∈ ℤ. The norm of a+bi is N(a+bi) = a²+b², which you may recognize as the squared distance from the origin. Equivalently, N(a+bi) = (a+bi)(a−bi) — the element times its complex conjugate. This gives a positive integer, and crucially, N is multiplicative: N((a+bi)(c+di)) = N(a+bi)·N(c+di). You can verify this directly, or notice it follows from |zw| = |z||w| for complex numbers.
In a general number field K = ℚ(α) of degree n, an algebraic integer α satisfies a degree-n polynomial over ℚ with n conjugates α = α₁, α₂, ..., αₙ (the roots). The norm of an element θ ∈ K is N(θ) = σ₁(θ)·σ₂(θ)···σₙ(θ), the product over all field embeddings σᵢ: K → ℂ. For K = ℚ(i), there are two embeddings: the identity and complex conjugation, giving N(a+bi) = (a+bi)(a−bi) = a²+b² — exactly what we computed above.
Multiplicativity N(αβ) = N(α)N(β) follows from the fact that each embedding is a ring homomorphism: σᵢ(αβ) = σᵢ(α)σᵢ(β), so the product over all embeddings factorizes. This multiplicativity is the norm's most useful property. It means if N(α) is a rational prime p, then α cannot factor as α = βγ with both β,γ non-units in O_K — otherwise N(β)·N(γ) = p with both factors integers greater than 1, a contradiction. So elements with prime norm are irreducible.
This gives a direct tool for Diophantine equations. To solve x²+y²=5 in integers, rewrite it as N(x+yi) = 5 in ℤ[i]. Since N(2+i) = 4+1 = 5, we find 2+i is a Gaussian prime, and the solutions to the Diophantine equation correspond to the Gaussian integer factorizations of 5. More generally, any question about which primes p are representable as x²+ny² translates into a question about factorization in ℤ[√(−n)], with the norm measuring whether factorization is possible.