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 simple criteria for the surjectivity of finite polynomial mappings KN → KN, particularly for real closed. The core of this article consists of criteria for the existence of K-rational points of finite  (commutative) The main tools are quadratic forms and their signatures (if K is an ordered field) which are derived from forms on such algebras, in particular from the trace and its generalizations. For finite polynomial mappings mapping degree is defined as such a signature. This mapping degre  will serve as a very effective tool to surjectivity of finite polynomial mappings over real closed fields K (as in differential topology for K = R ). it solves all the problems arising in Euler’s proof which will be  discussed in detail in the last section.