The problem is as follows.
Let $p$ is odd prime and suppose that a regular $p$-gon is constructible. Then show that $p$ is a Fermat prime.
I solved the $$\Bigg[\Bbb Q\left(\cos\frac{2\pi}{p}\right):\Bbb Q\Bigg]= \frac{p-1}{2}$$ and $$x^{2k+1} +1$$ is divisible by $x+1$ by using $p$th cyclotomic extension. And by the remark If $a$ is constructible number, $a$ is algebraic nubmer and $$[Q(a):Q]=2^r$$ for some $a \in N :$ national numbers
So i have the statement $$p=2^{r+1}+1.$$
But, i have to show $p$ is a Fermat prime. Then, how can i solve problem with my result above?
Please, help. (Better than recommendation for similar question, i want it to solve this condition.)
