---
tag: E45PA267S4
kind: concept
name: Transcendence basis
msc: [12F20]
sources:
  - url: https://en.wikipedia.org/wiki/Transcendental_extension
---

## Summary

A transcendence basis of a [[field extension|0EQWVHF3W5]] $L/K$ is a set $B$ of [[algebraically independent|KV4PT6KW10]] elements such that $L$ is [[algebraic|0PTBM7K0R2]] over $K(B)$.
It plays the role that a [[basis|0KCQ9Y3QKA]] plays for a vector space, with polynomial relations in place of linear ones.
[[Every extension has one|W635VBXEXG]], and all of them have the same cardinality, the [[transcendence degree|D7TR0YWR9H]].

## Explanations

### L1 human

A transcendence basis is a largest set of free variables inside a field extension.
Everything else in the big field is tied to these variables by polynomial equations.
In $\Q(x, \sqrt{x^3 + 1})$, the single element $x$ is a transcendence basis over $\Q$, since $y = \sqrt{x^3 + 1}$ satisfies $y^2 = x^3 + 1$.