Worked solution: Wantzel's algebraic impossibility proof via field extensions (1837)

Wantzel's criterion is strictly about straightedge-and-compass constructions in the classical Euclidean sense: an unmarked ruler used only to draw a line through two already-constructed points, and a compass used only to draw a circle through a constructed point centered at another. Relaxing either tool changes the answer. Archimedes (3rd century BCE) had already given a trisection using neusis — a marked ruler slid until two given lengths simultaneously align with two given curves — which effectively solves cubic equations directly and so bypasses the degree-$2^m$ restriction entirely.

In the same 1837 paper, Wantzel used an identical strategy (reduce to a cubic, show it is irreducible, note $3\neq2^m$) to prove that doubling the cube — constructing $\sqrt[3]{2}$ — is also impossible, and he additionally completed Gauss's theorem on which regular polygons are constructible. Charles Sturm apparently found improved proofs shortly afterward but never published them (Maths History, Wantzel biography).

Historically, this 1837 result stands among the earliest uses of field-degree arguments to prove a geometric impossibility, anticipating ideas Évariste Galois had developed a few years earlier that would soon crystallize into Galois theory. It converted three of antiquity's most famous open problems into settled, provable facts — though squaring the circle still needed Lindemann's transcendence proof of $\pi$, published 45 years later in 1882.