---
tag: 0M893X7YY7
kind: definition
name: Adjoining a root of a polynomial
msc: [12F05]
defines: 0NXWQK2Y56
uses: [0AWMPX35CW, 00X2A9G9W3, 0VFW4Y69BT]
sources:
  - url: https://en.wikipedia.org/wiki/Splitting_field
    locator: The field Ki[X]/(f(X))
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be [[irreducible|00X2A9G9W3]].
The field obtained by adjoining a root of $f$ to $K$ is the [[quotient ring|0VFW4Y69BT]] $K[x]/(f)$.
The map $K \to K[x]/(f)$, $c \mapsto c + (f)$, identifies $K$ with a subfield, and the class $\alpha = x + (f)$ is a root of $f$.

## Remarks

The quotient is a field, because [[$f$ is irreducible|0AWMPX35CW]].
Its degree over $K$ is $\deg f$, with the basis $1, \alpha, \dots, \alpha^{\deg f - 1}$.
For example, adjoining a root of $x^2 + 1$ to $\mathbb{R}$ gives a field isomorphic to $\mathbb{C}$.

## Explanations

### L1 human

You can create a root of a polynomial that has none.
Compute with polynomials in $x$, and replace $f(x)$ by zero each time it appears.
This is how $\mathbb{C}$ arises from $\mathbb{R}$ and $x^2 + 1$.