# Complex Polynomials

The Fundamental Theorem of Algebra states that the field of complex numbers is algebraically closed, implying that every polynomial equation of degree n has n roots within the complex numbers, with at least one being a solution where the polynomial evaluates to zero. Historically, the theorem’s origin traces back to the conjectures by Albert Girard in 1629 and René Descartes in 1637, though neither provided proofs including complex numbers. A first significant proof attempt was made by Jean d’Alembert (my personal hero) in 1746. The most notable advancements were made by Carl Friedrich Gauss, who, in 1799, offered a proof that, despite its gaps, marked a significant step forward in understanding the theorem. Gauss’s efforts were refined by Jean Robert Argand in 1814, who introduced an existence proof that set the stage for more concrete proofs, including those by Gauss himself in subsequent years using Euler’s earlier work. The theorem’s proof evolution highlights the gradual understanding and acceptance of complex numbers in solving polynomial equations, an understanding put further by Hellmuth Knesser (1940).

References

• Girard, A. (1629). Invention Nouvelle en l’Algèbre.
• Descartes, R. (1637). La Géométrie.
• d’Alembert, J. (1746). Recherches sur le calcul intégral.
• Gauss, C. F. (1799). Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse.
• Argand, J. R. (1814). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques.