An Alternative Proof of the Fundamental Theorem of Algebra

Darshan P.

Abstract

This note presents an alternative proof of the fundamental theorem of algebra. Specifically, we demonstrate that the degree of an irreducible polynomial in \( \mathbb{R}[X] \) is either 1 or 2, using a method that extends naturally to \( \mathbb{C}[X] \), proving that its irreducible polynomial degree is always 1.

1. Introduction

The goal of this paper is to revisit the fundamental theorem of algebra using a novel approach. The method applies to both \( \mathbb{R}[X] \) and \( \mathbb{C}[X] \).

2. Proof Setup

Let \( n > 1 \) and \( P \) be an irreducible polynomial in \( \mathbb{R}[X] \) of degree \( n \). Define \( n = 2 \) and consider the ideal \( \langle P \rangle \) generated by \( P \). The quotient ring \( \mathbb{R}[X] / \langle P \rangle \) forms a field. Define a mapping \( \psi : \mathbb{R}^n \to \mathbb{R}[X]/\langle P \rangle \):

\[\psi(a_0, a_1, \dots, a_{n-1}) = a_0 + a_1X + \cdots + a_{n-1}X^{n-1} + \langle P \rangle\]

This \( \psi \) is a group isomorphism and induces a field structure in \( \mathbb{R}^n \).

3. Norm Definition

Let \( \| \cdot \| \) be a norm on \( \mathbb{R}^n \) defined by:

\[\|x\| = \sup_{\|y\|=1} |x \cdot y|\]

Here, \( x \cdot y \) denotes the product in \( \mathbb{R}^n \). This norm satisfies \( \|1\| = 1 \) and \( \|x \cdot y\| \leq \|x\| \|y\| \).

4. Mathematical Derivations

Consider the following series:

\[\sum_{n=0}^{\infty} \frac{x^n}{n!}, \quad \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n}\]

Both series converge absolutely with respect to the defined norm. The product is commutative, and the exponential and logarithm functions are well-defined with:

\[\exp(\log(x)) = x \quad (\|x - 1\| < 1), \quad \log(\exp(x)) = x.\]

5. Topological Insights

For any \( x \in \mathbb{R}^n \) such that \( \| \exp(x) - 1 \| < 1 \):

The exponential function \( \exp \) is surjective, mapping \( \mathbb{R}^n \) onto \( \mathbb{R}^n \setminus \{0\} \). If \( G \) is the image of \( \exp \), then \( G \cdot x \subset \mathbb{R}^n \setminus G \).

\[\text{The complement of } G \text{ in } \mathbb{R}^n \setminus \{0\} \text{ is empty.}\]

Further analysis using topology and homology groups demonstrates that \( \ker(\exp) \) must be \( \{0\} \), proving that \( \mathbb{R}^n \setminus \{0\} \) is homeomorphic to \( \mathbb{R} \).

This leads to the conclusion that \( n = 2 \), completing the proof.

6. Conclusion

We have demonstrated an alternative proof of the fundamental theorem of algebra using the properties of the norm and the exponential/logarithmic functions in \( \mathbb{R}^n \). This method elegantly extends to \( \mathbb{C}[X] \).

References

  1. J. L. Lehman, Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic.
  2. Martin H. Weissman, An Illustrated Theory of Numbers, American Mathematical Society, 2017.