Part I, Chapter II — Analytical Summary¶
Impossibility of Cube Duplication and General Angle Trisection (§§ 1–3)
Synopsis of Arguments¶
Numerical Impossibility of the Delian Cube (§ 1): Doubling the unit cube requires $x^3 - 2 = 0$. Parity descent on $p^3 = 2q^3$ proves no rational root exists; for a cubic polynomial, absence of a linear factor is equivalent to irreducibility over $\mathbb{Q}$. Since $\deg(x^3 - 2) = 3 \neq 2^h$, the Delian problem is impossible with straightedge and compasses.
Function-Field Generalization via Domains of Rationality (§ 2): Adopting Kronecker’s domain of rationality $\mathbb{C}(\lambda)$ for a variable parameter $\lambda$, degree comparison on $[f(\lambda)]^3 = \lambda [\psi(\lambda)]^3$ ($3 \deg f = 1 + 3 \deg \psi$) proves $x^3 - \lambda = 0$ is irreducible over rational functions of $\lambda$, ruling out any universal straightedge-and-compass construction of $\sqrt[3]{\lambda}$.
Riemannian Monodromy Proof of Angle Trisection Impossibility (§ 3): Writing $\lambda = \cos\varphi + i\sin\varphi$, trisecting a general angle $\varphi$ is equivalent to solving $x^3 = \cos\varphi + i\sin\varphi$ over the field $\mathbb{Q}(\cos\varphi, \sin\varphi)$. Analytically continuing $\varphi \mapsto \varphi + 2\pi$ leaves every rational function of $(\cos\varphi, \sin\varphi)$ invariant while cyclically permuting the three roots $x_1 \to x_2 \to x_3 \to x_1$. Hence no root lies in $\mathbb{Q}(\cos\varphi, \sin\varphi)$, proving irreducibility of degree $3 \neq 2^h$—while cleanly distinguishing generic impossibility from special constructible angles such as $\varphi = \frac{\pi}{2}$.