---
tag: 00SW8B91E7
kind: definition
name: Transcendental element
msc: [12F20]
defines: 0NQMZ89X4A
uses: [0NPW1DWECM]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_element
    locator: Examples
---

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$.
The element $a$ is transcendental over $K$ if $a$ is not [[algebraic|0NPW1DWECM]] 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 $e$.