The Fundamental Theorem of Algebra

Abstract

The Fundamental Theorem of Algebra (FTA) asserts that every non-constant single-variable polynomial with complex coefficients possesses at least one complex root. This theorem serves as a cornerstone in mathematical discourse, bridging classical algebra and complex analysis. Despite its importance, the theorem admits various proofs rooted in different mathematical disciplines. This paper aims to provide a comprehensive exploration of the FTA by examining two distinct analytical approaches. First, we present a proof derived from advanced calculus, utilizing the extreme value theorem and properties of continuous functions on compact sets. Second, we examine a proof based on complex analysis, specifically leveraging Liouville’s Theorem. By delineating these methodologies, this study highlights the profound interplay between polynomial theory and higher-level mathematical analysis, offering a structured pedagogical framework for understanding the existence of roots in the complex plane.