Doubling the Cube by Compass and Straightedge Construction is Impossible

Theorem
There is no compass and straightedge construction to allow a cube to be constructed whose volume is double that of a given cube.

Proof
Suppose it is possible.

Then from a cube of edge length $L$ we can construct a new cube with edge length $\sqrt [3] 2 L$.

$\sqrt [3] 2$ is algebraic of degree $3$.

This contradicts Constructible Length with Compass and Straightedge.