👉 The proof of the fundamental theorem of algebra, which states that every non-constant polynomial equation with complex coefficients has at least one complex root, relies on complex analysis and topology. It begins by showing that a polynomial of degree n cannot remain bounded as its input grows to infinity in the complex plane; if it did, it would have no roots. By considering the polynomial's behavior at infinity and applying Liouville's theorem (which states that a bounded entire function must be constant), we deduce that the polynomial must have at least one root in the complex plane. This is further strengthened by showing that if a polynomial has no roots, it can be factored into linear terms, contradicting the assumption of no roots. Thus, every polynomial must have at least one complex root, with the full proof using tools from complex analysis and topology to ensure this conclusion holds universally.
