On Euler's proof of the Fundamental Theorem of Algebra

Abstract

In this article, Euler's (incomplete) proof of the Fundamental Theorem of Algebra from 1749 is used as a motivation to develop simple criteria for the surjectivity of finite polynomial mappings κN -→ κN, particularly for real closed fields κ. The core of this article consists of criteria for the existence of κ-rational points of finite (commutative) κ-algebras. The main tools are quadratic forms and their signatures (if κ is an ordered field) which are derived from κ-linear forms on such algebras, in particular from the trace and its generalizations. For finite polynomial mappings an algebraic mapping degree is defined as such a signature. This mapping degree will serve as a very effective tool to prove the surjectivity of finite polynomial mappings over real closed fields κ (as in differential topology for κ = r).addition, it solves all the problems arising in Euler's proof which will be discussed in detail in the last section.