Constructing regular polygons
October 19, 2011 5 Comments
Yesterday, as usual, we saw some cool stuff. The gist of it was that for polynomials that can’t be factored over , if and is a polynomial of degree , then . Here we can replace with any field , but for constructibility, we only care about .
What about constructing regular polygons? If I want to construct a regular -gon, I need to construct . So for which can I do this? I’m going to do something a little crazy. I’m going to consider the . This is a complex number, which you might recognize as an th root of unity. That is . What I want to know is the smallest polynomial which has as root. I’m not going to justify this, but if you take all the terms where is relatively prime to , and multiply them together, you get exactly this polynomial.
But the real question is the degree of this polynomial. It has one factor of for each relatively prime to . We have a name for that. It’s the Euler totient function . I have a prize for anyone who can tell me what the word “totient” means.
It turns out that by working over the complex numbers, I haven’t actually cheated (though I won’t justify this either), and the degree of . Well, we need to be a power of two if we have any chance of constructing it. Here are some facts about you might prove in an introductory number theory course:
- If is a prime dividing , then divides .
- If divides , then divides .
- If is odd, then .
- If is even, then .