The open mapping theorem and the fundamental theorem of algebra

TL;DR

This paper provides an elementary, fixed-point theorem-based proof of the open mapping theorem and the fundamental theorem of algebra, avoiding complex analysis and offering a new perspective on their connection.

Contribution

It introduces a novel, elementary proof of the open mapping theorem and derives the fundamental theorem of algebra using topological fixed point theorems, expanding the methods available for these classical results.

Findings

Elementary proof of the open mapping theorem using fixed point theorems

Derivation of the fundamental theorem of algebra via topological arguments

Generalization of the fundamental theorem of algebra

Abstract

This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological argument. The proof is elementary in nature and does not use any kind of integration (neither complex nor real). In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems. In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility to achieve it was questioned. We close this note with a simple generalization of the FTA. A short review of certain issues related to the OMT and the FTA is also included.