---
tag: 00X2A9G9W3
kind: definition
name: Irreducible polynomial
msc: ["12E05"]
defines: 03FKX7Z5DC
uses: [052SDEPZH7, 0Q9EZS90GY]
sources:
  - url: https://en.wikipedia.org/wiki/Irreducible_polynomial
    locator: Definition
---

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
A polynomial $f \in R[x]$ is irreducible over $R$ if it is an [[irreducible element|052SDEPZH7]] of the integral domain $R[x]$.
That is, $f$ is nonzero and not a unit, and if $f = g h$ with $g, h \in R[x]$, then $g$ or $h$ is a unit of $R[x]$.

## Remarks

Over a [[field|0QA8CHH8MG]] $K$, the units of $K[x]$ are the nonzero constants. So $f$ is irreducible if and only if $\deg f \ge 1$ and $f$ is not a product of two polynomials of smaller degree.
The polynomial $x^2 - 2$ is irreducible over $\mathbb{Q}$ but not over $\mathbb{R}$, where $x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2})$.
The polynomial $x^2 + 1$ is irreducible over $\mathbb{R}$ but not over $\mathbb{C}$.
Over $\mathbb{Z}$, the polynomial $2x + 2 = 2(x + 1)$ is not irreducible, because $2$ is not a unit of $\mathbb{Z}[x]$.

## Explanations

### L1 human

Irreducible polynomials are the prime numbers of polynomial rings.
Over a field, they are the polynomials that you cannot factor into polynomials of smaller degree.
Irreducibility depends on the field: $x^2 + 1$ factors over $\mathbb{C}$ but not over $\mathbb{R}$.