Every complex polynomial hides at least one complex root
The fundamental theorem of algebra says a non-constant polynomial with complex coefficients always has a complex root—and counted with multiplicity, exactly as many roots as its degree. Despite the name, proving it needs analysis or topology, not algebra alone.
Equivalently, the complex numbers form an algebraically closed field. Real-coefficient polynomials are included because reals sit inside the complexes. Successive division turns the "at least one root" form into the "exactly n roots with multiplicity" form. The name dates from an era when algebra meant equation theory; modern algebra does not rest on this theorem, and purely algebraic tools cannot prove it without transcendental input such as topology or complex analysis.
Anticipations abound. Peter Roth in 1608 suggested a degree-n equation may have n solutions; Albert Girard in 1629 claimed n solutions without demanding they be real. Leibniz in 1702 wrongly denied that x⁴ + a⁴ factors into real linears and quadratics; Euler corrected related mistakes by 1742. D'Alembert's 1746 attempt was incomplete and leaned on a result proved much later. Euler, de Foncenex, Lagrange, and Laplace assumed roots existed in some extension and tried to show they look like a + bi. Gauss's 1799 geometric proof had a topological gap filled by Ostrowski in 1920. Argand, an amateur, gave the first rigorous proof in 1806 (revised 1813) and stated the theorem for complex coefficients. Cauchy's 1821 textbook printed Argand's argument without credit.
None of those classical proofs were constructive. Weierstrass posed the constructive problem mid-nineteenth century and offered a solution in 1891 akin to Durand–Kerner plus homotopy continuation; Hellmuth Kneser (1940) and Martin Kneser (1981) refined constructive routes. Without countable choice one cannot constructively prove the usual statement for Dedekind reals, though reformulations exist. Corollaries include real factorizations into linears and quadratics only. The theorem guarantees roots; it does not hand you a formula to find them.
Source: Fundamental theorem of algebra
