Doubling the cube

Given the edge of a cube of side length $a$, construct in finitely many steps using only an idealized compass and an unmarked straightedge the edge length $x = \sqrt[3]{2}\,a$ of a second cube whose volume $x^3 = 2a^3$ is twice that of the given cube.

Proved impossible by Pierre Laurent Wantzel in 1837. Setting $a = 1$, doubling the cube requires constructing the real number $x = \sqrt[3]{2}$, which is a root of the polynomial $x^3 - 2 = 0$. By Eisenstein's criterion for the prime $p = 2$ (or rational root inspection), $x^3 - 2$ is irreducible over $\mathbb{Q}$, so the field extension $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]$ has degree $3$. Wantzel proved that any length constructible with an unmarked straightedge and compass lies in an iterated quadratic extension of $\mathbb{Q}$ of degree $2^k$; since $3$ does not divide $2^k$, $\sqrt[3]{2}$ is not constructible.