[Mathematical content warning: roots of unity]
Over the weekend I was reading and practicing with problems from Learning Modern Algebra. I came across these:
Let be a positive integer and let .
1. Establish the identity
2. If and are integers, show that
3. If and are integers, show that
I enjoyed thinking about these from a geometric perspective.
Most of the work of the first identity is being done by the definition of and the very notion of a root of unity. What are the solutions to ? to ? ? ?
Some quick facts, in case you’re bravely reading this extremely sketchy blog post without already being familiar with roots of unity:
- always has solutions, no matter what is.
- 1 is always a solution.
- -1 is often a solution.
- Besides for 1 and -1, all the solutions are non-real complex numbers.
- If you graph them in the complex plane, all the solutions are on the radius 1 circle away from (0,0).
- The points are always equally spaced from each other on that circle (which is all that $latex.
- Another way of saying the previous point is that provides one solution, and then the rest of the solutions are powers of .
- This guarantees that the solutions of will form a regular -gon.
A picture is worth a lot of words, via Wolfram MathWorld, and possibly violating their reuse permissions:
That first identity really is just summarizing the roots of unity situation. If all these powers of are the solutions of , then these s are the roots of and so each of those powers of is a factor of some polynomial that is equivalent to . Multiply all those factors together, and you get back the original polynomial:
The second identity is the one that I enjoyed thinking about as the result of scaling the entire unit circle by a factor of . So being on a circle of radius 1, the circle is now of radius . Which means that each of the roots has been multiplied by , so that the roots around the circle are now .
But if is now a root of the equation, instead of it must be , and likewise each the roots arranged around the circle will make when raised to the th power. And this is just what the second identity is saying:
The third identity is just what happens when you replace with , though this replacement is only relevant when the power of is odd and we get a negative (and so we end up swapping from subtraction to addition):
My textbook reports that the above factorization represents “an early attack on Fermat’s Last Theorem.” Feel free to link me to a readable explanation of what this early attack consisted in!
I find all of this so interesting, both mathematically and sort of meta-mathematically. I love the historical transition of algebra from “hey let’s solve some equations” to “hey here’s this entire metaphysics of structures whose names will give you absolutely no sense as to what they represent.” (I’m thinking of rings, ideals, modules, fields, domains, etc.) How did this happen? Where are the motivations?
Anyway, the above is a small part of that story.