Unique Factorization Domains

Introduction

The Fundamental Theorem of Arithmetic states that every integer (greater than 1) can be uniquely factored into a product of prime numbers. This property of unique factorization is one of the most important in number theory, and we can generalize it to abstract rings.

Definition

Definition 11.1: An integral domain $R$ is a Unique Factorization Domain (UFD) if every non-zero, non-unit element of $R$ can be written as a product of irreducible elements, and this factorization is unique up to the order of the factors and multiplication by units.

An element is irreducible if it cannot be factored into a product of two non-units. In a UFD, an element is irreducible if and only if it is prime (an element $p$ is prime if whenever $p$ divides $a b$ , then $p$ divides $a$ or $p$ divides $b$ ).

Key Properties

Existence of Factorization: Every non-zero, non-unit element can be written as a product of irreducibles.
Uniqueness of Factorization: This factorization is unique up to order and units.
Irreducible = Prime: In a UFD, an element is irreducible if and only if it is prime.

Relationship to Other Domains

The relationship between the classes of domains introduced so far is a beautiful hierarchy:

Euclidean Domains ⟹ Principal Ideal Domains ⟹ Unique Factorization Domains

Each implication is strict, meaning there are examples of PIDs that are not EDs, and UFDs that are not PIDs. For example, $Z [x]$ is a UFD but not a PID, as the ideal $(2, x)$ is not principal.

Examples

Example 1: The Ring of Integers

The ring $Z$ is a UFD. Every integer greater than 1 can be uniquely factored into a product of prime numbers. This is the Fundamental Theorem of Arithmetic.

Example 2: Polynomial Rings over Fields

For any field $F$ , the polynomial ring $F [x]$ is a UFD. This is a consequence of the fact that $F [x]$ is a Euclidean domain.

Example 3: Gaussian Integers

The ring of Gaussian integers $Z [i] = {a + b i ∣ a, b \in Z}$ is a UFD. This allows us to prove results about ordinary integers, such as Fermat's theorem on sums of two squares.

Example 4: A UFD that is not a PID

The ring $Z [x]$ is a UFD but not a PID. The ideal $(2, x)$ is not principal, showing that not every UFD is a PID.

Applications

Application 1: Number Theory

A classic application of UFDs is in number theory. The ring of Gaussian integers, $Z [i] = {a + b i ∣ a, b \in Z}$ , is a Euclidean domain, and therefore a UFD. Studying factorization in this ring allows one to prove results about ordinary integers, such as Fermat's theorem on sums of two squares, which states that an odd prime $p$ can be written as a sum of two squares if and only if $p \equiv 1 (\mod 4)$ .

Application 2: Algebraic Geometry

UFDs are important in algebraic geometry because they correspond to smooth varieties. The coordinate ring of a smooth affine variety is often a UFD.

Application 3: Cryptography

The unique factorization property is fundamental to many cryptographic protocols, including RSA, which relies on the difficulty of factoring large integers.