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Alternating Roots of Polynomials over Cayley-Dickson Algebras

Adam Chapman, Ilan Levin

Abstract

We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.

Alternating Roots of Polynomials over Cayley-Dickson Algebras

Abstract

We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.
Paper Structure (6 sections, 4 theorems, 20 equations)

This paper contains 6 sections, 4 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Cayley-Dickson Algebras
  3. Finding alternating roots
  4. Octave Code
  5. Correspondence theorem
  6. Generalized result

Key Result

Proposition 3.3

The intersection between the set of central roots of $p(N)$ and the image of the norm form $\operatorname{Norm} : A \rightarrow F$ is the set of norms of alternating roots of $f(x)$. For each such norm $N_0$ in this intersection, either all the elements in $A$ of this norm are alternating roots of $ is the one and only alternating root of $f(x)$ of norm $N_0$.

Theorems & Definitions (21)

  • Claim 3.1
  • proof
  • Claim 3.2
  • proof
  • Proposition 3.3
  • proof
  • Example 3.4
  • Remark 3.5
  • Remark 3.6
  • proof
  • ...and 11 more