Fundamental Theorem of Algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root, a result that, applied repeatedly, implies a polynomial of degree n has exactly n complex roots when counted with their multiplicity. Several eighteenth century mathematicians, including Jean le Rond d'Alembert in 1746, attempted proofs of the theorem, but each of these early attempts contained gaps or relied on unestablished assumptions. Carl Friedrich Gauss gave the first widely accepted proof in his 1799 doctoral dissertation, though his own original proof is now recognized to have contained a gap not fully closed until later in the nineteenth century, and Gauss himself later published three further, increasingly rigorous proofs of the theorem across his career. Despite its name, the theorem is now understood to belong more properly to complex analysis than to algebra in the modern sense, since every known proof relies on some analytic or topological property of the real or complex numbers, and the theorem is foundational to the study of polynomials because it guarantees the complex numbers are algebraically closed, meaning no polynomial equation with complex coefficients requires a broader number system to find its solutions.