---
tag: 0CQ357M0QS
kind: statement
role: theorem
name: Impossibility of doubling the cube
msc: [12F05]
uses: [04SD3F4FCR, 0SSSM1D475, 02BPKD7TMD, 0N5QW9K7MH]
sources:
  - url: https://en.wikipedia.org/wiki/Doubling_the_cube
    locator: Proof of impossibility
---

## Statement

The number $\sqrt[3]{2}$ is not a [[constructible number|0FMN2MN7TM]].
So the cube cannot be [[doubled|04SD3F4FCR]] with straightedge and compass.

## Remarks

The proof: $x^3 - 2$ is irreducible over $\mathbb{Q}$, by [[Eisenstein's criterion|02BPKD7TMD]] with $p = 2$. So $[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3$.
Since $3$ is not a power of $2$, the number $\sqrt[3]{2}$ is not constructible, by the [[degree theorem|0SSSM1D475]].

## History

Pierre Wantzel proved the impossibility in 1837.

## Explanations

### L1 human

The edge of the doubled cube is the cube root of $2$.
Its minimal polynomial has degree $3$, but constructible numbers have degree a power of $2$.