Engineering / Mathematics — Fields, Series and Factorisation
Unique Factorization Domains
Unique factorisation domains, the distinction between irreducible and prime, and the standard examples and counterexamples.
Executive summary
A unique factorisation domain is an integral domain where every non-zero non-unit factors into irreducibles uniquely up to order and units.
The condition is equivalent to every irreducible being prime, and its failure in some rings of algebraic integers is what motivated the development of ideal theory.
Learning objectives
- Define UFDs and state the equivalent characterisation.
- Distinguish irreducible from prime elements.
- Identify standard examples and failures.
01Definition and the key distinction
Irreducible and prime
A non-zero non-unit r is irreducible if r = ab forces a or b to be a unit.
It is prime if r | ab implies r | a or r | b.
Relationship
In any integral domain, prime implies irreducible. The converse holds precisely in unique factorisation domains.
The distinction is invisible in the integers because the two coincide there, which is why elementary treatments use the words interchangeably. In general they are different conditions and unique factorisation is exactly their agreement.
02The standard failure
In Z[√−5], the element 6 has two genuinely different factorisations.
6 = 2 · 3 = (1 + √−5)(1 − √−5)All four factors are irreducible, verified by a norm argument: the norm is multiplicative and takes values 4, 9, 6 and 6, and no element has norm 2 or 3, so none of the factors splits further. No two of the factors are associates, since the only units are ±1.
The failure is precisely that 2 is irreducible but not prime: it divides the product on the right without dividing either factor.
03The hierarchy
- Euclidean domain
Division with remainderZ, F[X], F[[X]], Z[i] - Principal ideal domain
Every ideal generated by one elementAll Euclidean domains, plus others - Unique factorisation domain
Irreducibles are primeAll PIDs, plus Z[X] and F[X,Y] - Integral domain
No zero divisorsAll of the above, plus Z[√−5]
| Ring | Euclidean | PID | UFD |
|---|---|---|---|
| Z | Yes | Yes | Yes |
| F[X] | Yes | Yes | Yes |
| Z[i] | Yes | Yes | Yes |
| Z[X] | No | No | Yes |
| F[X, Y] | No | No | Yes |
| Z[√−5] | No | No | No |
Each implication is strict, and the table gives a witness for each strictness. The Euclidean property is the one that yields algorithms, which is why the algorithmic content of this collection lives in the top row.
04Frequently asked questions
Why does prime always imply irreducible?
Because if a prime r factors as ab, then r divides ab, hence divides one factor, say a. Writing a = rc gives r = rcb, so cb = 1 and b is a unit. The converse needs the ring to be a UFD.
Does unique factorisation imply a gcd exists?
Yes — take the minimum exponent of each irreducible across the two factorisations. What may fail in a UFD that is not a PID is that the gcd need not be an integer combination of the arguments, so Bezout is unavailable.
How is factorisation restored in Z[√−5]?
By factoring ideals rather than elements. The ideal generated by 6 factors uniquely into prime ideals, and the two element factorisations correspond to two different ways of grouping those prime ideals into principal ones.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 383-390.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.