---
tag: KV4PT6KW10
kind: concept
name: Algebraic independence
msc: [12F20]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_independence
same_as:
  wikidata: Q1495342
  wikipedia: Algebraic independence
---

## Summary

A set of elements of a field extension is algebraically independent if no nonzero polynomial relation with coefficients in the base field holds among them.
It is the analogue for polynomials of [[linear independence|0CRMGPAH83]] for linear combinations.
A single element is algebraically independent exactly when it is [[transcendental|0NQMZ89X4A]].
Maximal algebraically independent sets are [[transcendence bases|E45PA267S4]].

## Explanations

### L1 human

Two numbers are algebraically dependent when some polynomial equation ties them together.
For example, $\pi$ and $\pi^2$ are dependent over $\Q$, since $y - x^2 = 0$ at $x = \pi$, $y = \pi^2$.
In the field $\Q(x, y)$ of rational functions in two variables, $x$ and $y$ are independent: no nonzero polynomial in them is zero.
Whether $\pi$ and $e$ are algebraically independent over $\Q$ is not known.