Impossibility of trisecting the angle

Statement

The angle $\pi/3$, that is, $60^\circ$, cannot be trisected with straightedge and compass. Equivalently, $\cos(\pi/9)$ is not a constructible number.

Remarks

The proof: $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$ shows that $c = \cos(\pi/9)$ is a root of $8x^3 - 6x - 1$. Then $u = 2c$ is a root of $x^3 - 3x - 1$. The polynomial $x^3 - 3x - 1$ has no rational root, by the rational root theorem. So it is irreducible, and $[\mathbb{Q}(c) : \mathbb{Q}] = 3$. Since $3$ is not a power of $2$, the number $c$ is not constructible. The angle $\pi/3$ itself is constructible, so no general trisection method exists.

History

Pierre Wantzel published the proof in 1837.

Explanations

L1 human

If you could trisect every angle, you could trisect $60^\circ$ and construct an angle of $20^\circ$. The cosine of $20^\circ$ has degree $3$ over the rationals, so it is not constructible.