---
tag: 0D6KH8Y59N
kind: statement
role: theorem
name: Polynomial ring over a UFD is a UFD
msc: [13F15]
uses: [06YMXJMXYS, 0QFV5Y1Z5X, 0Q9EZS90GY]
sources:
  - url: https://en.wikipedia.org/wiki/Unique_factorization_domain
    locator: Examples
---

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is a unique factorisation domain.

## Remarks

The proof uses [[Gauss's lemma|0QFV5Y1Z5X]] and unique factorisation in $K[x]$, where $K$ is the field of fractions of $R$.
The irreducible elements of $R[x]$ are the irreducible elements of $R$ and the primitive polynomials of positive degree that are irreducible in $K[x]$.
By induction, $R[x_1, \dots, x_n]$ is a unique factorisation domain. In particular, $\mathbb{Z}[x]$ and $K[x, y]$ are.
These rings are not [[principal ideal domains|0RZYDVZ475]]: $(2, x)$ in $\mathbb{Z}[x]$ and $(x, y)$ in $K[x, y]$ are not principal.

## Explanations

### L1 human

Integer polynomials factor uniquely into irreducible pieces, like integers do.
The pieces are prime numbers and irreducible primitive polynomials.
For example, $2x^2 - 2 = 2 (x - 1)(x + 1)$.