= Impossibility of doubling the cube by straightedge and compass
{title2=$[\mathbb Q(\sqrt[3]2):\mathbb Q]=3$}
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.
Back to article page
