---
tag: 0MB4TGMXG6
kind: definition
name: Algebraic number
msc: [11R04]
defines: 0MNF0BYFQC
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_number
    locator: lead section
---

## Definition

A complex number $\alpha$ is algebraic if there is a nonzero polynomial $f$ with rational coefficients such that $f(\alpha) = 0$.
The degree of $\alpha$ is the smallest degree of such a polynomial.

## Remarks

One may use integer coefficients instead, by clearing denominators.
Examples are each rational number (degree $1$), $\sqrt{2}$ and $i$ (degree $2$), and $\sqrt[3]{2}$ (degree $3$).
The algebraic numbers form a field, and this field is countable.
A complex number that is not algebraic is [[transcendental|079EGSD32Y]].

## Explanations

### L1 human

An algebraic number is a solution of a polynomial equation with integer coefficients.
For example, $\sqrt{2}$ is algebraic because it solves $x^2 - 2 = 0$.