Comment by zengid 3 days ago
This is really interesting.. does anyone who knows more about Gauss's proof know why you can construct a 5 sided polygon with ruler and compass, but not a 7 or 11 sided polygon? Why do some primes work and others not?
I can give a very fast and incomplete explanation, but you must trust me.
In the complex numbers, the vertices of a pentagon are z^5-1=0. You can factorize it as (z^4+z^3+z^2+z+1)*(z-1)=0. The hard part is solving z^4+z^3+z^2+z+1=0.
Now that equation can't be factorized, and has degree 4. It's important that the solutions have a property that is related to the degree of the equation so they have a property that is 4.
With a compass and a straightedge you can solve only equations of degree 2, that is like taking a square root. If you repeat the process you can solve (some) equations of degree 4. So after a few tricks, you can solve the equation and draw the pentagon.
For 17, the equation is z^16+z^15+...+z+1=0. So the property is 16 and you must use the square root a few times. Each time the solutions double their property, so you get 1 -> 2 -> 4 -> 8 -> 16. Near the bottom of the article is the formula, and it's possible to see a lot of nested and repeated square roots.
For 7, the equation is z^6+z^5+...+z+1=0. So the property of the solutions is 6. With the square root you can only double the property, so you get 1 -> 2 -> 4 -> 8 -> 16 -> 32 ... but you can never get a solution which has a property equal to 6.
(There are more technicals details. You can solve some equations of degree 16, for example to draw the 17agon, but you can't solve every one of them.)
If you are interested and have the time you can watch the 2 videos from the YouTube channel Another Roof[1] on that subject. He spends some time on easy stuff too to allow the general public to relatively understand the bases, so don't be surprised if the videos are quite long.
This was downvoted. I guess someone though it was a joke. But it's part of the correct answer. More details in https://en.wikipedia.org/wiki/Fermat_number
This is described in quite an approachable way in the video I posted in another comment. https://www.youtube.com/watch?v=EX7U0DGBmbM
cyclotomic polynomials and Galois theory
a 17-gon reduces to a 4th-degree polynomial, and a 2nd degree one, which can be solved in radicals, by studying the permutations and multiplicities of the roots of this polynomial, in which the solutions are multiples of n*pi/17.
For 17, Gauss noticed that cos(360°/17) can be written only with elementary operations, see https://www.heise.de/imgs/18/2/1/2/3/3/6/4/siebzehneck-b95b5...
Later he proved that all n-gons with $n=2^k*p_1…*p_r$ where the p_i are Fermat-primes (2^(2^m)+1 prime, today we only know of 3, 5, 17, 257, 65537) are constructible. The opposite direction, i.e. all other n are not constructible, was only a few years later proved. Look up "Theorem of Gauss-Wantzel". I only skimmed the proof, but it seems to generalize the concept of constructing the cos of the angle with "Galois-Theory".
(edit: or see https://en.wikipedia.org/wiki/Constructible_polygon)