Roots of Polynomials

Abstract

In olden times, when algebraic theorems were scanty, the following statement received the title of the Fundamental Theorem of Algebra: “A given polynomial of degree n with complex coefficients has exactly n roots (multiplicities counted).” The first to formulate this statement was Alber de Girard in 1629, but he did not even try to prove it. The first to realize the necessity of proving the Fundamental Theorem of Algebra was d’Alembert. His proof (1746) was not, however, considered convincing. Euler (1749), Faunsenet (1759) and Lagrange (1771) offered their proofs but these proofs were not without blemishes, either. The first to give a satisfactory proof of the Fundamental Theorem of Algebra was Gauss. He gave three different versions of the proof (1799, 1815 and 1816) and in 1845 he additionally published a refined version of his first proof. For a review of the different proofs of the Fundamental Theorem of Algebra, see [Ti]. We confine ourselves to one proof. This proof is based on the following Rouché’s theorem, which is of interest by itself.