COMPUTABLE IMPLEMENTATION OF ``FUNDAMENTAL THEOREM OF ALGEBRA"
Jon A. Sjogren, X. Li, Minghui Zhao, C. Lu · International Journal of Pure and Apllied Mathematics · 2013
The Fundamental Theorem of Algebra (FTA) has been studied for more than 300 years: more or less satisfactory proofs of FTA emerged in the 18th and 19th centuries.Proofs denoted as 'algebraic' or 'elementary' derived from the axioms defining a Real-Closed Field (RCF).A proof is given that brings up-to-date work of Gauss (1816) and P. Gordan (1879).It does not refer explicitly to the complex numbers but instead works with auxiliary polynomials in two variables.We report that computer software has been developed to effect symbolic calculation in the context of exact arithmetic.Some examples show how these routines apply to the algebra of symmetric multinomial forms used in Laplace's proof (1795) of FTA, as well as to the theory of Sylvester forms and the Bézoutian formulation of the resultant.