Algebraic element

Definition

Let $L/K$ be a field extension 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 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.

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 $\mathbb{Q}$.