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Quantifier Elimination and Invariant Theory: Applications to Quaternions, Octonions, and Other Algebras

Maximilian Illmer

Abstract

We build on our previous paper~\cite{constructive} by using the general method introduced there in conjunction with invariant theory. This yields quantifier elimination results for the classical quaternions, octonions, as well as other classes of finite-dimensional algebras over real closed and algebraically closed fields. In particular, the first two examples answer an open question posed recently in~\cite{savi}.

Quantifier Elimination and Invariant Theory: Applications to Quaternions, Octonions, and Other Algebras

Abstract

We build on our previous paper~\cite{constructive} by using the general method introduced there in conjunction with invariant theory. This yields quantifier elimination results for the classical quaternions, octonions, as well as other classes of finite-dimensional algebras over real closed and algebraically closed fields. In particular, the first two examples answer an open question posed recently in~\cite{savi}.
Paper Structure (7 sections, 13 theorems, 56 equations)

This paper contains 7 sections, 13 theorems, 56 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Introduction
  3. Preliminaries from Algebra
  4. Preliminaries from Logic
  5. Linear Algebraic Groups and Invariant Theory
  6. Results
  7. Acknowledgements

Key Result

Theorem 1.1

Let $A$ be a structure in the language $\mathcal{L}$ and denote by $\mathrm{Aut}(A)$ its automorphism group. Then for $m$-tuples $\underline{a},\underline{b}\in A^m$ one has

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • ...and 19 more