 Solution

A constructible regular polygon can be translated, rotated, and rescaled by constructible operations to one inscribed in the unit circle with a vertex at $(1,0)$. Thus the real and imaginary parts of $\zeta =e^{2\pi i/p}$ are constructible. Part (c) puts both in a real field $K$ of degree $2^n$ over $\mathbb{Q}$. Consequently $\zeta \in K(i)$, and $[K(i):\mathbb{Q}]=2^{n+1}$ because $K\subset \mathbb{R}$.

For prime $p$, the supplied irreducibility of $1+x+\cdots +x^{p-1}$ shows that $[\mathbb{Q}(\zeta ):\mathbb{Q}]=p-1$. The intermediate-field tower law implies that $p-1$ divides $2^{n+1}$. Every positive divisor of a power of two is a power of two, soThis necessary condition includes $p=2$ with $r=0$. It does not assert sufficiency for arbitrary integers of this form.