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Who first gave a complete proof of the fundamental theorem of algebra?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

The first complete proof of the fundamental theorem of algebra was provided by the German mathematician Carl Friedrich Gauss in his doctoral dissertation of 1799.12

While other mathematicians such as d’Alembert, Euler, and Lagrange had previously worked on the theorem, their efforts were hindered by a lack of rigorous foundations for polynomials and number systems. Gauss’s work was significant because he tackled the problem of existence directly, using methods from calculus and geometry to demonstrate that a solution must exist. This achievement marked a shift toward the rigorous, internal development of mathematics. Gauss later provided three additional proofs of the theorem throughout his career.12