Unique Factorization Domains — Mathematics Knowledge System

Overview

A unique factorization domain (UFD) is an integral domain in which every nonzero non-unit element factors into irreducible elements, and that factorization is unique up to reordering and multiplication by units — exactly generalizing the Fundamental Theorem of Arithmetic for ℤ. The ring of integers ℤ and any polynomial ring F[x] over a field F are UFDs, and Gauss's lemma shows that if R is a UFD, so is the polynomial ring R[x]. Not every integral domain is a UFD, however: the classic counterexample is ℤ[√-5], where 6 factors two genuinely different ways into irreducibles.

Intuition

In ℤ, 12 factors as 2²×3 and there's no other way to break it into primes (aside from writing −2 instead of 2 alongside a sign flip) — that predictable, unique breakdown is what a UFD guarantees in general. The famous failure ℤ[√-5] shows this isn't automatic: 6 = 2×3 = (1+√-5)(1-√-5), and all four factors 2, 3, 1+√-5, 1-√-5 are irreducible (none factors further) yet genuinely different, so unique factorization collapses. This is exactly the phenomenon that motivated Kummer and Dedekind to invent 'ideal numbers' (ideals) to restore a unique factorization at the level of ideals even when elements misbehave.

Formal Definition

Definition

An integral domain R is a UFD if every nonzero non-unit r ∈ R can be written as a product of irreducibles, uniquely up to order and unit multiples:

r = u \cdot p_1 p_2 \cdots p_k, \quad u \text{ a unit},\ p_i \text{ irreducible}

Existence of factorization

p_1 \cdots p_k = q_1 \cdots q_m \implies k=m \text{ and } p_i = u_i q_{\sigma(i)} \text{ for units } u_i, \text{ permutation } \sigma

Uniqueness up to units and reordering

\text{PID} \implies \text{UFD}

Every principal ideal domain is a UFD

R \text{ UFD} \implies R[x] \text{ UFD}

Gauss's Lemma consequence: polynomial rings over a UFD are UFDs

Worked Examples

60 = 2² × 3 × 5, using the standard prime factorization.
$60 = 2^2 \times 3 \times 5$
Any other factorization into irreducibles (primes, up to sign) is just a reordering or sign-flip of these same factors, e.g. 60 = (−2)²×3×5 is 'the same' factorization since units are ±1.
$60 = (-2)(-2)(3)(5)$

Answer: 60 = 2²·3·5, unique up to reordering and the unit ±1.

Practice Problems

Difficulty 5/10

Why is every field trivially a UFD?

Difficulty 6/10

State the relationship between PIDs, UFDs, and integral domains, and give an example of a UFD that is not a PID.

Difficulty 6/10

In ℤ[√-5], compute N(2), N(3), N(1+√-5), N(1-√-5) where N(a+b√-5)=a²+5b², and use multiplicativity of the norm to argue 2 is irreducible.

Quiz

Summary

A UFD is an integral domain where every nonzero non-unit factors uniquely (up to order/units) into irreducibles, generalizing the Fundamental Theorem of Arithmetic.
Every PID is a UFD, and if R is a UFD then so is R[x] (Gauss's Lemma); not every UFD is a PID (e.g. ℤ[x]).
ℤ[√-5] is the classic non-UFD example: 6 = 2·3 = (1+√-5)(1-√-5) are genuinely different irreducible factorizations.

References

WebsiteWikipedia — Unique factorization domain