Worked solution: The Gauss–Wantzel theorem: constructible regular polygons via cyclotomic fields

A regular $n$-gon is constructible with straightedge and compass exactly when the primitive $n$-th root of unity $\zeta_n=e^{2\pi i/n}$ is a constructible complex number, equivalently when $\cos(2\pi/n)$ is constructible. The minimal polynomial of $\zeta_n$ over $\mathbb{Q}$ is the $n$-th cyclotomic polynomial, of degree $\varphi(n)$ (Euler's totient function, counting integers from $1$ to $n$ coprime to $n$), so $[\mathbb{Q}(\zeta_n):\mathbb{Q}]=\varphi(n)$.

The argument used for $n=17$ generalizes without change: whenever $\varphi(n)$ is a power of $2$, the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^\times$ (which controls how the roots of unity permute among themselves) has an order that is a power of $2$, and groups of order $2^s$ always have a chain of subgroups each half the size of the last. This chain of subgroups produces exactly the kind of nested periods used in Steps 2–3, giving a tower of quadratic extensions from $\mathbb{Q}$ up to $\mathbb{Q}(\zeta_n)$, and hence constructibility.

So the question 'which regular polygons are constructible?' reduces entirely to a question about a single arithmetic function: for which $n$ is $\varphi(n)$ a power of $2$? The next step answers this by factoring $\varphi(n)$.