Explore the logical challenges surrounding roots of unity in complex algebra through this 29-minute video lecture. Examine the rational parametrization of the unit circle and the differences between specifying unit quadrance complex numbers using rational expressions versus trigonometric functions. Investigate the difficulties in defining angles for complex numbers and the use of infinite series for circular functions. Delve into the construction of regular polygons, particularly the seven-gon, and consider purely algebraic approaches to solving equations like z^7=1 without relying on infinite processes. Gain insights into the foundational issues underlying these common algebraic objects and their geometric representations.