Transcendental element

Definition

Let $L/K$ be a field extension and let $a \in L$. The element $a$ is transcendental over $K$ if $a$ is not algebraic over $K$. That is, $f(a) \neq 0$ for every nonzero $f \in K[x]$.

Remarks

The number $\pi$ is transcendental over $\mathbb{Q}$, but it is algebraic over $\mathbb{R}$, as a root of $x - \pi$. The variable $x$ is transcendental over $K$ in the field $K(x)$ of rational functions. If $a$ is transcendental over $K$, then $K(a)$ is isomorphic to $K(x)$, and $[K(a) : K]$ is infinite.

Explanations

L1 human

A transcendental element satisfies no polynomial equation over the smaller field. It behaves like a free variable. The transcendental numbers are the complex numbers that are transcendental over $\mathbb{Q}$, such as $\pi$ and $e$.