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

Write $n = 2^k p_1^{e_1} \cdots p_m^{e_m}$ for distinct odd primes $p_i$. Since Euler's totient function is multiplicative, $\varphi(n) = 2^{k-1}\prod_{i=1}^m p_i^{e_i - 1}(p_i - 1)$ for $k\ge1$ (drop the leading factor if $k=0$). For $\varphi(n)$ to be a power of $2$, every factor $p_i^{e_i-1}(p_i-1)$ must itself be a power of $2$.

Since $p_i$ is odd, the term $p_i^{e_i-1}$ is a power of $2$ only when $e_i-1=0$, i.e. $e_i=1$: each odd prime factor of $n$ must occur exactly once. What remains is that $p_i - 1$ itself be a power of $2$, say $p_i - 1 = 2^{r_i}$, so $p_i = 2^{r_i}+1$. A short argument (if $r_i$ had an odd factor $d>1$, then $2^{r_i}+1$ would be divisible by $2^{r_i/d}+1$, a proper factor) shows such $p_i$ can be prime only when $r_i$ is itself a power of $2$, $r_i=2^{s_i}$, giving $p_i = 2^{2^{s_i}}+1$ — a Fermat prime.

Combining both conditions: $n$ is constructible exactly when $n = 2^k p_1 \cdots p_m$ for distinct Fermat primes $p_i$ (Gauss 1801, Art. 366). This is precisely the pattern behind $17=2^{2^2}+1$, the third Fermat prime, and behind $n=3,5$, the first two.