The fundamental theorem of algebra: A survey of history and proofs

The fundamental theorem of algebra states that every polynomial of positive degree with complex coefficients has at least one zero. This paper details the history, gives biographical information on mathematicians whose work led to the proofs, and gives proofs from different branches of mathematics.;The mathematicians of ancient civilizations could solve some equations. Many old manuscripts show how to solve special cases. These ancients were unable to deduce general formulas for solving equations.;Algebra begins its evolution in the 13th century. Around the 16th century algebraic symbolism takes form. Mathematicians are unclear about negative and complex numbers. Around 1600, Albert Girard states the fundamental theorem of algebra for the first time. Throughout the 17th and 18th century, many mathematicians attempted to prove it.;In 1799, Carl Gauss states and proves the fundamental theorem of algebra in his dissertation. Throughout his lifetime he gives three more proofs.;The easiest to understand proofs are from the field of Complex Variables. A. L. Cauchy is responsible for many of the theorems and concepts.;E. Galois lived in the early 1800's and his work led to Galois Theory. This theory is used in proving the fundamental theorem using Algebra.;The Cauchy-Riemann equations lead to a proof using Analysis.;Topology had its beginnings in the 19th century, but much of the development has taken place in the 20th century. The proof of the fundamental theorem of algebra using topology is based on the fundamental group of the circle.;Although the proof of the theorem was not given until 1799, mathematicians were sure of its correctness. They invented approximation methods to locate the roots which they knew existed.