Cyclotomic
Roots of unity. Using $e^{i \theta} = \cos(\theta) + i \sin(\theta)$, and connecting them to trigonometry, etc. Features a ton of problems from Math Prize for Girls. Despite the name of the unit, cyclotomic polynomials themselves don't appear very often; the idea of roots of unity is much more prevalent.
Unit Overview
A unit featuring trigonometry, Roots of Unity, and a bit of vectors. Contrary to the name, Cyclotomic Polynomials don't appear much, rather, roots of unity is much more significant. A big idea presented in this unit is the duality between complex numbers and trigonometry - most algebraic trigonometry problems can be turned into complex numbers and vice versa. This unit is around an average B unit in difficulty, but probably on the harder spectrum of the average range.
Notable Problems
- Gauss Sum: An involved example problem with a bit of number theory mixed in.
- Putnam 2015/A3: An important problem that introduces how substitutions with roots of unity make a problem much easier.
- MP4G 2016/20 : A problem that really makes you think about what roots of unity do in this scenario, featuring combo.
- IMO 1990/6 and MP4G 2017/19: Very important problems that help you find the connection between vectors and roots of unity.
Advice
This unit is pretty helpful for computational contests and has a little less relevance to olympiad problems. It is beneficial for anyone aiming to get better at late AIME algebra.