---
tag: D7TR0YWR9H
kind: concept
name: Transcendence degree
msc: [12F20]
sources:
  - url: https://en.wikipedia.org/wiki/Transcendental_extension
---

## Summary

The transcendence degree of a [[field extension|0EQWVHF3W5]] $L/K$ is the cardinality of a [[transcendence basis|E45PA267S4]].
It counts the algebraically independent elements needed so that the rest is algebraic.
It does not depend on the choice of basis.
It is the field-theoretic notion of dimension: the function field of an irreducible variety has transcendence degree equal to the dimension of the variety.

## Explanations

### L1 human

The transcendence degree counts how many free variables a field extension has.
The field $\C(x, y)$ has transcendence degree $2$ over $\C$, like the plane.
The field of rational functions on a curve has transcendence degree $1$, like a line.
An extension has transcendence degree $0$ exactly when it is algebraic.