Algebra, fundamental theorem of

The theorem that states that any polynomial with complex coefficients has a root in the field of complex numbers. The theorem was first stated by A. Girard in 1629 and by R. Descartes in 1637 in a formulation different from the one employed today. C. MacLaurin and L. Euler made the formulation more precise and gave it a form which is equivalent to the one in use today: Any polynomial with real coefficients can be decomposed into a product of linear and quadratic factors with real coefficients. A proof of the fundamental theorem of algebra was first given by J. d'Alembert in 1746. There followed the proofs of Euler, P.S. Laplace, J.L. Lagrange and others in the second half of the 18th century. All these proofs are based on the assumption that some "ideal" roots of the polynomial in fact exist, after which it is demonstrated that at least one of them is a complex number. C.F. Gauss was the first to prove the fundamental theorem of algebra without basing himself on the assumption that the roots do in fact exist. His proof essentially consists of constructing the splitting field of a polynomial. All proofs of the theorem involve some form of topological properties of real and complex numbers. The role of topology has ultimately been reduced to the single assumption that a polynomial of odd degree with real coefficients has a real root.

References

[1]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)
[2]	S. Lang, "Algebra" , Addison-Wesley (1974)
[3]	I.G. Bashmakova, "On a proof of the fundamental theorem of algebra" Istor. Mat. Issled. : 10 (1957) pp. 257–304 (In Russian)

Comments

For a proof based on the Brouwer fixed-point theorem cf. [a1].

References

[a1]	B.H. Arnold, "A topological proof of the fundamental theorem of algebra" Amer. Math. Monthly , 56 (1949) pp. 465–466