---
tag: CYCJ46W6P6
kind: definition
name: Transcendence basis
msc: [12F20]
defines: E45PA267S4
uses: [0WQ1A8SHTZ, KV4PT6KW10, 09EYMKMSB9, 4V81611P9N]
sources:
  - url: https://en.wikipedia.org/wiki/Transcendental_extension
---

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
A subset $B \subseteq L$ is a *transcendence basis* of $L/K$ if:

1. $B$ is [[algebraically independent|KV4PT6KW10]] over $K$;
2. $L$ is an [[algebraic extension|09EYMKMSB9]] of the subfield $K(B)$ generated by $K$ and $B$.

## Remarks

Equivalently, $B$ is a maximal algebraically independent subset of $L$.
Example: in the [[field of rational functions|4V81611P9N]] $K(x_1, \dots, x_n)$, the set $\set{x_1, \dots, x_n}$ is a transcendence basis over $K$, and so is $\set{x_1^2, x_2, \dots, x_n}$.
Non-example: $\set{x_1}$ is not a transcendence basis of $K(x_1, x_2)/K$, because $x_2$ is not algebraic over $K(x_1)$, so clause 2 fails.
For an algebraic extension, the empty set is the unique transcendence basis.