List of Unique Factorisation Domains
Introduction
This article lists the unique factorisation domains of the corpus, with the irreducibles of each example, and it records the failure of unique factorisation for each non-example with the two factorisations that witness it. Every entry points to the article that introduces the object.
A unique factorisation domain is an integral domain in which every nonzero nonunit is a product of irreducibles and the factorisation is unique up to order and associates; equivalently, by the criterion of Unique Factorisation Domains, every irreducible element is prime. The list gathers the Euclidean domains and their rings of integers, the polynomial rings over a field or a unique factorisation domain, and the power series rings, and it records $\mathbb{Z}[\sqrt{-5}]$ and the imaginary quadratic rings of class number greater than one on the other side.
The article introduces nothing and proves nothing. It records examples and non-examples side by side.
The Criterion and the Transfer
The equivalence of unique factorisation with "every irreducible is prime" is the working criterion of Unique Factorisation Domains, and Euclid's lemma is its first consequence. The transfer to a polynomial ring is Gauss's lemma: if $R$ is a unique factorisation domain then $R[x]$ is one, with the content $c(f)$ and the primitive polynomials carrying the arithmetic. Eisenstein's criterion is the standard test for an irreducible polynomial. All of this is the subject of Unique Factorisation Domains; a catalogue records only which rings the corpus meets.
| Statement | Content | Introduced in |
|---|---|---|
| a domain is a UFD iff every irreducible is prime | irreducibles and primes | Unique Factorisation Domains |
| $R$ a UFD implies $R[x]$ a UFD | content and Gauss's lemma | Unique Factorisation Domains |
| Eisenstein's criterion | irreducibility of a polynomial over a UFD | Unique Factorisation Domains |
The Unique Factorisation Domains and Their Irreducibles
| Unique factorisation domain | The irreducibles, up to associates | Introduced in |
|---|---|---|
| $\mathbb{Z}$ | the rational primes $\pm p$ | The Integers |
| $\mathbb{Z}[i]$ | $1+i$, the rational primes $p \equiv 3 \bmod 4$, and the elements $\pi$ with $N(\pi) = p$ for $p \equiv 1 \bmod 4$ | Examples of Rings and Fields |
| $\mathbb{Z}[\sqrt{-2}]$ | $\sqrt{-2}$, and the elements of norm $p$ for a rational prime $p$ with $-2$ a square modulo $p$ | Examples of Rings and Fields |
| $\mathbb{Z}[\sqrt{2}]$ | $\sqrt{2}$, and the elements of norm $\pm p$ | Examples of Rings and Fields |
| $\mathbb{Z}[\tfrac{1+\sqrt{-3}}{2}]$, the Eisenstein integers | $\sqrt{-3}$, and the elements of norm $p \equiv 1 \bmod 3$ | Examples of Rings and Fields |
| $k[x]$ | the monic irreducible polynomials; the linear ones when $k$ is algebraically closed | Polynomial Rings and Rational Functions |
| $\mathbb{F}_q[x]$ | the monic irreducible polynomials over the finite field | Finite Fields |
| $\mathbb{Z}[x]$ | the rational primes $p$, and the primitive irreducible polynomials | Polynomial Rings and Rational Functions |
| $k[x_1, \dots, x_n]$ | the irreducible polynomials in $n$ variables | Polynomial Rings and Rational Functions |
| $k[x_1, x_2, \ldots]$ in infinitely many variables | the irreducible polynomials in finitely many variables | Examples of Rings and Fields |
| $k[[x]]$ | $x$ alone, up to units | Examples of Rings and Fields |
| every field, for instance $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{F}_p$, $\mathbb{Q}_p$ | none: every nonzero element is a unit | Fields |
| $\mathcal{O}_{K,\mathrm{P}}$, a discrete valuation ring | the uniformiser, a generator of the maximal ideal | Dedekind Domains and Ideal Class Groups |
| $\mathbb{Q}[x]$, $\mathbb{R}[x]$, $\mathbb{C}[x]$ | the monic irreducibles; the linear ones over $\mathbb{C}$ | Polynomial Rings and Rational Functions |
The Gaussian integers give the standard worked example: $2 = -i(1+i)^2$ ramifies, an odd prime $p \equiv 3 \bmod 4$ remains irreducible, and a prime $p \equiv 1 \bmod 4$ splits as $\pi\bar\pi$ with $N(\pi) = p$, the case $5 = (2+i)(2-i)$. In the power series ring the only irreducible is $x$, and the localisation $\mathbb{Z}_{(p)}$ has the single irreducible $p$; a discrete valuation ring has one irreducible up to associates, and this is recorded in List of Local Rings and Valuations.
Unique Factorisation in the Local and Power Series Rings
The discrete valuation rings are the unique factorisation domains with the smallest possible supply of irreducibles: one, up to associates, because the nonzero ideals form a single chain.
| Unique factorisation domain | The single irreducible | The maximal ideal it generates | Introduced in |
|---|---|---|---|
| $\mathbb{Z}_{(p)}$ | $p$ | $p\mathbb{Z}_{(p)}$ | Localization and the Fraction Field |
| $\mathbb{Z}_p$ | $p$ | $p\mathbb{Z}_p$ | Absolute Values, Valuations and Completions |
| $k[t]_{(t)}$ | $t$ | $tk[t]_{(t)}$ | Absolute Values, Valuations and Completions |
| $\mathbb{F}_p[x]_{(x)}$ | $x$ | $x\mathbb{F}_p[x]_{(x)}$ | Examples of Rings and Fields |
| $k[[x]]$ | $x$ | $(x)$ | Examples of Rings and Fields |
| $\mathcal{O}_{K,\mathrm{P}}$ | a generator of $\mathrm{P}$ | $\mathrm{P}$ | Dedekind Domains and Ideal Class Groups |
Each of these rings is a principal ideal domain and hence a unique factorisation domain, and in each the factorisation of an element is the statement that its value is a nonnegative integer. The power series ring is not a field, and its single irreducible is a nonunit that is not a zero divisor.
Domains That Are Not Unique Factorisation Domains
| Domain | The failure | Class number | Introduced in |
|---|---|---|---|
| $\mathbb{Z}[\sqrt{-5}]$ | $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$, four pairwise nonassociate irreducibles | $2$ | Examples of Rings and Fields |
| $\mathbb{Z}[\sqrt{-6}]$ | the class group is nontrivial, so some element has two factorisations | $2$ | Dedekind Domains and Ideal Class Groups |
| $\mathbb{Z}[\sqrt{-10}]$ | the class group is nontrivial | $2$ | Dedekind Domains and Ideal Class Groups |
| $\mathbb{Z}[\sqrt{-14}]$ | the class group is cyclic of order $4$ | $4$ | Dedekind Domains and Ideal Class Groups |
| $\mathcal{O}_K$ with nontrivial class group | unique factorisation of elements fails; ideals still factor uniquely | varies | Dedekind Domains and Ideal Class Groups |
| $\overline{\mathbb{Z}}$ | Bézout and not Noetherian, hence not a UFD | — | Bézout Domains |
For $\mathbb{Z}[\sqrt{-5}]$ the element $2$ is irreducible but not prime, since $\mathbb{Z}[\sqrt{-5}]/(2)$ is not a domain; the ideal class group has order $2$, generated by the class of the non-principal prime $P = (2, 1+\sqrt{-5})$, and the failure of unique factorisation of elements is repaired by the unique factorisation of ideals. The four factors $2$, $3$, $1+\sqrt{-5}$ and $1-\sqrt{-5}$ all have norm $4$, $9$, $6$, $6$, and no element of norm $2$ or $3$ exists in the ring, which is why they are irreducible; the units are $\pm 1$, which is why they are pairwise nonassociate. This computation is recorded in Dedekind Domains and Ideal Class Groups and in Examples of Rings and Fields, and it is the failure that Unique Factorisation Domains treats in detail.
Unique Factorisation That Survives Without Principal Ideals
A unique factorisation domain need not be a principal ideal domain, and the standard witnesses are polynomial rings.
| Ring | Why it is a UFD | Why it is not principal | Introduced in |
|---|---|---|---|
| $\mathbb{Z}[x]$ | Gauss's lemma applied to $\mathbb{Z}$ | the ideal $(2, x)$ is not principal | Polynomial Rings and Rational Functions |
| $k[x_1, \dots, x_n]$, $n \geq 2$ | the iterated Gauss's lemma | the ideal $(x_1, x_2)$ is not principal | Polynomial Rings and Rational Functions |
| $k[x_1, x_2, \ldots]$ | the iterated Gauss's lemma on finitely many variables | not Noetherian | Examples of Rings and Fields |
| $R[x]$ for $R$ a UFD | Gauss's lemma | not principal in general | Polynomial Rings and Rational Functions |
| $\mathbb{F}_q[x, y]$ | the iterated Gauss's lemma over $\mathbb{F}_q$ | the ideal $(x, y)$ is not principal | Polynomial Rings and Rational Functions |
| $\mathbb{Z}[x_1, \dots, x_n]$ | Gauss's lemma applied to $\mathbb{Z}$, iterated | the ideal $(2, x_1)$ is not principal | Polynomial Rings and Rational Functions |
The first two are the reason the ladder of List of Structures from Rings to Fields has a strict step between a unique factorisation domain and a principal ideal domain, and the third is the reason a unique factorisation domain need not be Noetherian.
Warnings
| Object | Why it is not a unique factorisation domain of this list | Introduced in |
|---|---|---|
| $M_2(\mathbb{R})$, $\mathbb{H}$, $\mathbb{B}$, $\mathbb{H}_{\mathbb{D}}$ | not commutative, and the matrix, biquaternion and split-biquaternion rings have zero divisors | Matrix Algebras, Quaternion Algebra, Biquaternion Algebra, Split-Biquaternion Algebra |
| $\mathbb{D}$, $\mathbb{D}'$ | not domains: the split-complex numbers have $e_+e_- = 0$ and the dual numbers a nilpotent | Split-Complex Algebra, Dual-Numbers Algebra |
| $\mathbb{Z}/6\mathbb{Z}$ | not a domain: $2 \cdot 3 = 0$ | Modular Arithmetic and the Ring of Residues |
| $\mathbb{O}$ | not a ring | Octonion Algebra |
| a general Dedekind domain $\mathcal{O}_K$ | a UFD exactly when its class number is $1$ | Dedekind Domains and Ideal Class Groups |
Summary
This article has listed the unique factorisation domains of the corpus with the irreducibles of each: the rational primes in $\mathbb{Z}$, the Gaussian, quadratic and Eisenstein primes in the rings of integers of class number one, the monic irreducible polynomials in $k[x]$ and $\mathbb{F}_q[x]$, the primitive irreducibles and the rational primes in $\mathbb{Z}[x]$, the irreducible polynomials in several and in infinitely many variables, the single irreducible $x$ in $k[[x]]$, and no irreducible at all in a field. It has recorded the failures: the two factorisations of $6$ in $\mathbb{Z}[\sqrt{-5}]$, the nontrivial class groups of $\mathbb{Z}[\sqrt{-6}]$, $\mathbb{Z}[\sqrt{-10}]$, $\mathbb{Z}[\sqrt{-14}]$ and a general $\mathcal{O}_K$, and the non-Noetherian Bézout domain $\overline{\mathbb{Z}}$.
Summary of Notation
A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.
| Symbol | Meaning |
|---|---|
| $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{F}_q$ | The number systems and the finite fields |
| $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{-2}]$, $\mathbb{Z}[\sqrt{2}]$, $\mathbb{Z}[\sqrt{-5}]$ | Quadratic integer rings |
| $\mathbb{Z}[\tfrac{1+\sqrt{-3}}{2}]$ | The Eisenstein integers |
| $\mathbb{Z}[\sqrt{-6}]$, $\mathbb{Z}[\sqrt{-10}]$, $\mathbb{Z}[\sqrt{-14}]$ | Imaginary quadratic rings of class number $> 1$ |
| $\overline{\mathbb{Z}}$, $\mathcal{O}_K$ | All algebraic integers, ring of integers of $K$ |
| $N(\cdot)$, $c(f)$ | Norm, content of a polynomial |
| $k[x]$, $k[x_1,\dots,x_n]$, $\mathbb{Z}[x]$, $k[[x]]$ | Polynomial and power series rings |
| $\mathbb{Q}_p$, $\mathbb{Z}_{(p)}$ | $p$-adic numbers, localisation at $p$ |
| $\mathbb{H}$, $\mathbb{B}$, $\mathbb{H}_{\mathbb{D}}$, $M_2(\mathbb{R})$ | Non-commutative rings, named in the warnings |
Further Reading
- Paulo Ribenboim, Classical Theory of Algebraic Numbers (Springer, 2001), for the quadratic rings, their class numbers and the explicit factorisations.
- Oscar Zariski and Pierre Samuel, Commutative Algebra, Volume I (Van Nostrand, 1958), for unique factorisation, Gauss's lemma and content.
- Irving Kaplansky, Commutative Rings (University of Chicago Press, revised ed. 1974), for unique factorisation domains and their failure in Dedekind domains.