---
tag: 0NPW1DWECM
kind: definition
name: Algebraic element
msc: [12F05]
defines: 0H628RSGWX
uses: [0WQ1A8SHTZ, 00EHX6FWC0, 0Q9EZS90GY]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_element
    locator: lead section
---

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$.
The element $a$ is algebraic over $K$ if there is a nonzero polynomial $f \in K[x]$ with $f(a) = 0$.

## Remarks

The number $\sqrt{2}$ is algebraic over $\mathbb{Q}$, because it is a [[root|00EHX6FWC0]] of $x^2 - 2$.
Every element $a \in K$ is algebraic over $K$, as a root of $x - a$.
The element $a$ is algebraic over $K$ if and only if $[K(a) : K]$ is finite.
An element that is not algebraic is [[transcendental|00SW8B91E7]].

## Explanations

### L1 human

An algebraic element solves a polynomial equation with coefficients in the smaller field.
The algebraic numbers are the complex numbers that are algebraic over $\mathbb{Q}$.