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An elementary algebraic proof of the fundamental theorem of algebra

Katelyn S. Clark, Pace P. Nielsen

TL;DR

This paper introduces a new elementary algebraic proof of the fundamental theorem of algebra by exploiting long division and a universal ring construction $R_{f,k}$ that adjoins monic factors of a polynomial of degree $k$. It shows a precise dimension formula $ ext{dim}_F(R_{f,k})=inom{n}{k} ext{dim}_F(R)$ and uses this, together with a 2-adic argument, to establish a generalized FTA in the spirit of Shipman, avoiding group-theoretic tools. The real numbers are connected to the complex closure by demonstrating the required square-root closure and root-existence conditions, yielding $ar{ ext{R}}= ext{C}$. The work also discusses computational aspects (e.g., Gröbner bases) and how the approach can be taught in a first course, highlighting its algebraic nature and potential extensions.

Abstract

We give a new proof of the fundamental theorem of algebra. It is entirely elementary, focused on using long division to its fullest extent. Further, the method quickly recovers a more general version of the theorem recently obtained by Joseph Shipman.

An elementary algebraic proof of the fundamental theorem of algebra

TL;DR

This paper introduces a new elementary algebraic proof of the fundamental theorem of algebra by exploiting long division and a universal ring construction that adjoins monic factors of a polynomial of degree . It shows a precise dimension formula and uses this, together with a 2-adic argument, to establish a generalized FTA in the spirit of Shipman, avoiding group-theoretic tools. The real numbers are connected to the complex closure by demonstrating the required square-root closure and root-existence conditions, yielding . The work also discusses computational aspects (e.g., Gröbner bases) and how the approach can be taught in a first course, highlighting its algebraic nature and potential extensions.

Abstract

We give a new proof of the fundamental theorem of algebra. It is entirely elementary, focused on using long division to its fullest extent. Further, the method quickly recovers a more general version of the theorem recently obtained by Joseph Shipman.

Paper Structure

This paper contains 6 sections, 5 theorems, 21 equations.

Table of Contents

  1. Introduction and history
  2. Learning what we can from long division
  3. Proving the fundamental theorem of algebra
  4. Connecting to the real numbers
  5. A few final remarks
  6. Acknowledgements

Key Result

Proposition 2.4

Let $R$ be a nonzero ring. If $f(x)\in R[x]$ is monic, then $R_{f,1}$ is a free $R$-module of rank $n:=\deg(f)$.

Theorems & Definitions (10)

  • Proposition 2.4
  • proof
  • Theorem 2.6
  • proof
  • Theorem 2.9
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof