Sufficient Conditions for Unconstructibility of a number

Question

What are some sufficient conditions for a regular polygon to not be constructible via ruler and compass? I posted my own answer below, but I am curious to see if there are any other non-trivial sufficient conditions for a number to not be constructible.

Answer 1

Proposition: Let $P$ be a regular $n$-gon, where $n\geq3$ is prime and $n-1$ $\ne 2^k$ for some natural number $k\geq2$. Then the $n$-gon is not constructible with ruler and compass.

Proof: To show this, we will demonstrate that $\beta=e^{2\pi i/n}$ is not constructible. We know that $\beta$ is a root of the minimal polynomial $x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$, but clearly $\beta\ne 1$. Therefore, $\beta$ is a root of $x^{n-1}+x^{n-2}+...+x+1$. Now, by a corollary to Eisenstein's criterion, we know that cyclotomic polynomials are irreducible over $\mathbb Q$. Therefore, since the minimal polynomial of $\beta$ over $\mathbb Q$ must have degree equal to a power of $2$ in order for $\beta$ to be constructible and $[\mathbb Q(\beta): \mathbb Q]$ = $n-1\ne 2^k$, it follows that $\beta$ is not constructible.