---
tag: 12JQCGTQ8W
kind: concept
name: Transcendental extension
msc: [12F20]
specializes: [0EQWVHF3W5]
sources:
  - url: https://en.wikipedia.org/wiki/Transcendental_extension
same_as:
  wikidata: Q3733413
  wikipedia: Transcendental extension
---

## Summary

A transcendental extension is a [[field extension|0EQWVHF3W5]] that is not [[algebraic|0PTBM7K0R2]].
So it contains at least one [[transcendental element|0NQMZ89X4A]], an element that is a root of no nonzero polynomial over the base field.
The size of such an extension is measured by its [[transcendence degree|D7TR0YWR9H]], the number of independent transcendental elements it needs.
Fields of rational functions and function fields of varieties are the main examples.

## Explanations

### L1 human

An algebraic extension adds roots of polynomials, such as $\sqrt 2$ added to $\Q$.
A transcendental extension adds something that solves no polynomial equation, such as $\pi$ added to $\Q$, or a free variable $t$.
The field $\Q(t)$ of rational functions in $t$ is the simplest example: no nonzero polynomial with rational coefficients vanishes at $t$.