Visualizing Cyclotomic Polynomials

This is about Cyclotomic Polynomials and an interesting way to view them visually.
Part 2: • Visualizing Cyclotomic...
Another Follow-up: • Modifiers of Cyclotomi...
There are some unjustified jumps in logic within the video. I did that to keep the length down - a video that covers every detail would be less entertaining - but I do have detailed notes written to support these 'claims'.

[@The9thDoctor]

Great video! Earned a spot in the math playlist for sure. I hope more people find this channel.

[@wargreymon2024]

This channel is gold

[@luisalejandrohernandezmaya254]

Excelent video.
I have a question maybe you can answer it.
The cyclotomic polynomial when n=p = prime is
Φ_p(x)=x^{p-1}+x^{p-2}+...+x+1
I have to prove Φ_p is irreducible. Hint: Use Eisenstain criterion.
The problem is that all a_k = 1 so there is not a prime q such that q|a_k for all k

[@TheGrayCuber]

Take (x + 1) as a an input of Φ_p, then you can use the binomial theorem to get primes

[@luisalejandrohernandezmaya254]

@@TheGrayCuber Oh! Thank you so much!! God bless you!

[@area51xi]

I need a little more hand holding on this. Got the overall feel but need to go back a notch and review some more fundamentals of nth roots and number theory. Thanks.

[@pawebielinski4903]

You do realise, however, that cyclotomic polynomials can in fact have coefficients other than --1, 0, 1?

[@TheGrayCuber]

Yes, those cases are more complicated to break down into regular polygons, but for example with 105 there are two points that are part of both a pentagon and a triangle

[@pawebielinski4903]

@@TheGrayCuber I see, I was curious what you'd do in that case.