---
tag: 052SDEPZH7
kind: definition
name: Irreducible element
msc: [13A05]
defines: 0YMT0D2442
uses: [09S3PFBEPM, 019NFGFPR5]
sources:
  - url: https://en.wikipedia.org/wiki/Irreducible_element
    locator: Definition in an integral domain
---

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
An element $p \in R$ is irreducible if it has these three properties.

1. $p \neq 0$.
2. $p$ is not a [[unit|019NFGFPR5]].
3. If $p = a b$ with $a, b \in R$, then $a$ is a unit or $b$ is a unit.

## Remarks

The irreducible elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ prime.
In $\mathbb{Z}[i]$, the number $3$ is irreducible, but $5 = (2 + i)(2 - i)$ is not.
Every [[prime element|0MNHXBSX3M]] is irreducible. The converse [[can fail|0CHE3J7QMX]].
For polynomials over a field, the irreducible elements are the [[irreducible polynomials|00X2A9G9W3]].

## Explanations

### L1 human

An irreducible element cannot be split into smaller factors.
The only factorizations are trivial ones, with a unit as one factor, like $7 = (-1)(-7)$.
In the integers, the irreducible elements are the primes and their negatives.