Algebraic number

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.

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 $x^2 - 2 = 0$.