Impossibility of doubling the cube by straightedge and compass ($[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$)

Doubling the volume of a unit cube requires constructing its new side length $\sqrt[3]{2}$. Its minimal polynomial $X^3-2$ is irreducible by Eisenstein's criterion at two, so this length has degree three. Every constructible number belongs to a finite tower of quadratic extensions, whose total degree is a power of two. The tower law for field extensions would force the degree three to divide that power of two, a contradiction. Thus no such straightedge and compass construction exists.