---
tag: G5PVJMM37Z
kind: concept
name: Purely transcendental extension
msc: [12F20]
specializes: [12JQCGTQ8W]
sources:
  - url: https://en.wikipedia.org/wiki/Transcendental_extension
---

## Summary

A purely transcendental extension of a field $K$ is an extension generated by [[algebraically independent|KV4PT6KW10]] elements.
It is isomorphic to a [[field of rational functions|4V81611P9N]] over $K$ in some set of variables.
Every field extension is an algebraic extension of a purely transcendental one, so these extensions are the building blocks of [[transcendental extensions|12JQCGTQ8W]].

## Explanations

### L1 human

A purely transcendental extension adds free variables and nothing else.
The field $\Q(x, y)$ of fractions of polynomials in $x$ and $y$ is the model example.
Adding $\sqrt 2$ as well gives an extension that is no longer purely transcendental, since $\sqrt 2$ is tied to $\Q$ by the equation $t^2 = 2$.