---
tag: HMF8G4EBDX
kind: definition
name: Transcendental extension
msc: [12F20]
defines: 12JQCGTQ8W
uses: [0WQ1A8SHTZ, 09EYMKMSB9, 00SW8B91E7, 079EGSD32Y]
sources:
  - url: https://en.wikipedia.org/wiki/Transcendental_extension
---

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
The extension $L/K$ is *transcendental* if it is not [[algebraic|09EYMKMSB9]], that is, if some element of $L$ is [[transcendental|00SW8B91E7]] over $K$.

## Remarks

Example: $\R/\Q$ is transcendental, because $\pi$ is a [[transcendental number|079EGSD32Y]].
Example: for a field $K$, the extension $K(t)/K$ of rational functions is transcendental, since $g(t) \ne 0$ for every nonzero $g \in K[x]$.
Non-example: $\Q(\sqrt 2)/\Q$ is not transcendental, because it is finite and hence algebraic.
Every field extension $L/K$ is the composite of a purely transcendental step $K \subseteq K(B)$ and an algebraic step $K(B) \subseteq L$, where $B$ is a transcendence basis.