= General angle quintisection is impossible with straightedge and compass
{title2=$96c^5-120c^3+30c-5=0$}
Consider the constructible angle whose cosine is $5/6$ and sine is $\sqrt{11}/6$. If one fifth of this angle were constructible, its cosine $c$ would satisfy the displayed polynomial, by the identity $\cos5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta$. The identity follows by taking the real part of $(\cos\theta+i\sin\theta)^5$ and substituting $\sin^2\theta=1-\cos^2\theta$.
The polynomial is irreducible by <Eisenstein's criterion> at five, so $c$ has degree five over the rational numbers. The initial angle field $\mathbb Q(\sqrt{11})$ already has degree two, and adjoining any finite collection of <constructible numbers> gives total degree over the rational numbers a power of two. Such a field cannot contain the degree-five element $c$, by the <tower law for field extensions>. Hence this constructible angle cannot be divided into five equal angles by <straightedge and compass construction>. Some particular angles can be quintisected; the obstruction disproves a construction valid for all angles.
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