Explore a critical examination of the Fundamental Theorem of Algebra in this 28-minute mathematics lecture. Delve into restatements of the theorem that aim for meaningful and correct formulations, avoiding assumptions about continuity and complex numbers. Learn about rational complex arithmetic and the concept of quadrance for complex numbers. Discover the potential for significant mathematical contributions in this area, as the lecturer emphasizes the need for new, deeper understanding and computationally-oriented proofs. Gain insights into the importance of finding approximate zeroes to a prescribed level of accuracy and understand why resolving this problem could be as significant as major 20th-century mathematical achievements.