---
tag: 0DFYTNZ6AM
kind: statement
role: proposition
name: Finite extensions are algebraic
msc: [12F05]
uses: [0NRYPM9Y27, 09EYMKMSB9]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_extension
    locator: Some properties
---

## Statement

Let $L/K$ be a [[finite extension|0NRYPM9Y27]].
Then $L/K$ is [[algebraic|09EYMKMSB9]].

## Remarks

The proof: let $n = [L : K]$ and $a \in L$. The $n + 1$ elements $1, a, \dots, a^n$ are linearly dependent over $K$. A linear dependence $c_0 + c_1 a + \dots + c_n a^n = 0$ gives a nonzero polynomial with root $a$.
So every element of $L$ has degree at most $[L : K]$ over $K$.
The converse is false, as the algebraic numbers show.
An extension generated by finitely many algebraic elements is finite, by the [[tower law|0A7KJ68XHT]].

## Explanations

### L1 human

In a finite-dimensional extension, the powers of an element cannot all be independent.
A dependence between the powers is a polynomial equation for the element.