Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On polynomial equations over split-octonions: the arbitrary field case

Artem Lopatin

TL;DR

This work resolves polynomial equations over split-octonions $\mathbf{O}$ with scalar polynomial coefficients and a possibly non-scalar constant term, over an arbitrary field $\mathbb{F}$. The authors reduce solving $f(x)=c$ to solving equations in $\mathbb{F}$ by introducing Generalized Fibonacci polynomials and expressing powers in $\mathbf{O}$ via $\operatorname{tr}(a)$ and $n(a)$, producing explicit solution sets parameterized by $(\lambda,\mu)$. They extend prior results from algebraically closed fields to the general field setting and provide constructive canonical forms and explicit root descriptions for $x^2=c$ and $x^3=c$, including real-case formulas. The results deepen the understanding of polynomial equations in nonassociative algebras and have potential implications for physics and related mathematical frameworks employing split-octonions.

Abstract

Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.

On polynomial equations over split-octonions: the arbitrary field case

TL;DR

This work resolves polynomial equations over split-octonions with scalar polynomial coefficients and a possibly non-scalar constant term, over an arbitrary field . The authors reduce solving to solving equations in by introducing Generalized Fibonacci polynomials and expressing powers in via and , producing explicit solution sets parameterized by . They extend prior results from algebraically closed fields to the general field setting and provide constructive canonical forms and explicit root descriptions for and , including real-case formulas. The results deepen the understanding of polynomial equations in nonassociative algebras and have potential implications for physics and related mathematical frameworks employing split-octonions.

Abstract

Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.
Paper Structure (9 sections, 8 theorems, 57 equations)

This paper contains 9 sections, 8 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Equations over octonions
  3. Results
  4. Octonions
  5. Split-octonions
  6. The group $\mathop{\rm Aut}(\mathbf{O})$
  7. Canonical octonions
  8. Polynomial equations
  9. Roots of octonions

Key Result

Proposition 2.2

A minimal set of representatives for the $\mathop{\rm Aut}(\mathbf{O})$-orbits in $\mathbf{O}$ consists of the following elements: where $\alpha,\beta \in \mathbb{F}$. In other words, $\mathbf{O}$ is the disjoint union of the following $\mathop{\rm Aut}(\mathbf{O})$-orbits: with $\alpha,\beta \in \mathbb{F}$.

Theorems & Definitions (17)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Corollary 3.5
  • ...and 7 more