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

Trisecting a given angle $\theta$ with straightedge and compass means producing an angle $\theta/3$ from the given one by a finite sequence of legal Euclidean moves. To make 'legal move' checkable, Wantzel recasts the whole question arithmetically: in coordinates, constructing an angle amounts to constructing its cosine (dropping a perpendicular to the unit circle recovers $\theta$ from $\cos\theta$, and conversely), so trisecting $\theta$ is possible exactly when $\cos(\theta/3)$ is a constructible real number.

The choice of $\theta = 60^\circ$ is the economical counterexample: $\cos 60^\circ = 1/2$ is constructible, so $60^\circ$ itself can be drawn — and it then suffices to show that $20^\circ$, equivalently $x = 2\cos 20^\circ$, cannot be. A single unconstructible instance kills the general claim, since any claimed universal trisection procedure would have to work on this angle in particular.

Wantzel's 1837 memoir in Liouville's journal develops the plan in three algebraic moves: (1) every straightedge-and-compass step enlarges the field of available coordinates by at most a quadratic extension, so constructible numbers have degree a power of $2$ over $\mathbb{Q}$ (Steps 1–2); (2) the triple-angle formula turns $\cos 20^\circ$ into a root of the cubic $x^3 - 3x - 1 = 0$ (Step 3); (3) that cubic is irreducible over $\mathbb{Q}$ (Step 4), giving degree $3$ — not a power of $2$ (Step 5).