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arxiv: 2404.04976 · v6 · submitted 2024-04-07 · 🧮 math.AG · math.LO· math.RA

On the first-order theories of quaternions and octonions

Enrico Savi This is my paper

Pith reviewed 2026-05-24 02:16 UTC · model grok-4.3

classification 🧮 math.AG math.LOmath.RA
keywords quaternionsoctonionsfirst-order theoriesreal closed fieldsmodel completenessquantifier eliminationalgebraic setsZariski topology
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The pith

The first-order ring theories of quaternions and octonions are axiomatized and their models are exactly the algebras over real closed fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies a first-order axiomatization in the language of rings for both the quaternions and the octonions. It then proves that any model of these theories is isomorphic to a quaternion algebra or an octonion algebra built over some real closed field. The two theories are bi-interpreted with the theory of real closed fields, which immediately yields completeness and model completeness while ruling out quantifier elimination. Model completeness is applied to the zero sets of ordered polynomials (the slice-regular functions) and is used to introduce algebraic sets together with a Zariski topology on these algebras.

Core claim

We provide an axiomatization of the L-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field, respectively. We bi-interpret these theories in terms of real closed fields and we prove they are complete, model complete and they do not have quantifier elimination. Then, we focus on the class of ordered polynomials. Over H and O these polynomials are of special interest in hypercomplex analysis since they are slice regular. We deduce some fundamental properties of their zero loci from model completeness and we introduce the notions of algebraic sets and Zariski topology. Finally, we 1)

What carries the argument

Bi-interpretation of the ring theories of the quaternions and octonions with the theory of real closed fields.

If this is right

  • The theories are complete and model complete.
  • The theories lack quantifier elimination.
  • Model completeness implies fundamental properties of the zero loci of ordered polynomials.
  • Algebraic sets and a Zariski topology can be defined on the algebras.
  • Quantifier elimination fails for the fragment of ordered formulas, and the family of algebraic sets is completely characterized.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Model-theoretic transfer from real closed fields may supply new decidability or definability results for hypercomplex analysis.
  • The same axiomatization technique could be tested on other normed division algebras or non-associative structures.
  • The Zariski topology defined via model completeness offers a candidate for algebraic geometry over quaternions and octonions that is independent of any choice of coordinates.

Load-bearing premise

The quaternions and octonions are interpreted strictly in the language of rings, without any added order or other symbols.

What would settle it

A model of the stated axioms that is not isomorphic to any quaternion or octonion algebra over a real closed field, or conversely an algebra over a real closed field that fails one of the axioms.

read the original abstract

Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field, respectively. We bi-interpret these theories in terms of real closed fields and we prove they are complete, model complete and they do not have quantifier elimination. Then, we focus on the class of ordered polynomials. Over $\mathbb{H}$ and $\mathbb{O}$ these polynomials are of special interest in hypercomplex analysis since they are slice regular. We deduce some fundamental properties of their zero loci from model completeness and we introduce the notions of algebraic sets and Zariski topology. Finally, we prove the failure of quantifier elimination for the fragment of ordered formulas and we completely characterize the family of algebraic sets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper axiomatizes the first-order L-theories (L = language of rings) of the quaternions ℍ and octonions 𝕆. It characterizes the models of these theories, up to isomorphism, as quaternion and octonion algebras over real closed fields. The theories are shown to be bi-interpretable with the theory of real closed fields (RCF), from which the authors deduce completeness and model completeness (but not quantifier elimination). The second half studies ordered polynomials over ℍ and 𝕆 (noting their relation to slice-regular functions), derives properties of zero loci from model completeness, introduces algebraic sets and a Zariski topology, proves failure of quantifier elimination already for the ordered fragment, and completely characterizes the algebraic sets.

Significance. If the axiomatization and bi-interpretation results hold, the work supplies the first model-theoretic description of these division algebras in the pure ring language and transfers key properties (completeness, model completeness) from RCF. The application to zero loci of ordered polynomials and the resulting algebraic geometry over ℍ/𝕆 is a natural and potentially useful extension to hypercomplex analysis. The explicit failure of quantifier elimination for ordered formulas is a clear negative result that sharpens the positive model-completeness statement.

major comments (2)
  1. [§3] §3 (bi-interpretation): the claim that the center and the ordering are definable from the ring language alone is load-bearing for the transfer of model completeness from RCF; the manuscript should exhibit the explicit formulas (or at least the quantifier complexity) used to define the center and the positive cone inside the ring language.
  2. [Theorem 5.4] Theorem 5.4 (characterization of algebraic sets): the proof that every algebraic set is a finite union of zero loci of ordered polynomials relies on model completeness; if the model-completeness argument in §3 only yields model completeness after adding constants for a real closed field, the reduction to the pure ring language needs an additional uniformity argument that is not spelled out.
minor comments (3)
  1. Notation: the symbol “L” is used both for the ring language and later for ordered formulas; a distinct symbol for the ordered fragment would improve readability.
  2. [Abstract] The abstract states that the theories “do not have quantifier elimination,” but the body only proves failure for the ordered fragment; the statement should be qualified accordingly.
  3. [Introduction] Reference to the classical Artin–Schreier theory of real closed fields is missing; adding a sentence in the introduction would situate the bi-interpretation result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments, which will help improve the clarity of the manuscript. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (bi-interpretation): the claim that the center and the ordering are definable from the ring language alone is load-bearing for the transfer of model completeness from RCF; the manuscript should exhibit the explicit formulas (or at least the quantifier complexity) used to define the center and the positive cone inside the ring language.

    Authors: We agree that explicit formulas will strengthen the exposition. The center is defined by the quantifier-free formula ∀y (xy = yx). The positive cone on the center is the set of elements that are sums of squares in a manner compatible with the norm form of the division algebra; this is expressible by a first-order formula whose quantifier complexity we will record explicitly. In the revised manuscript we will insert these formulas (with their quantifier prefixes) into §3. revision: yes

  2. Referee: [Theorem 5.4] Theorem 5.4 (characterization of algebraic sets): the proof that every algebraic set is a finite union of zero loci of ordered polynomials relies on model completeness; if the model-completeness argument in §3 only yields model completeness after adding constants for a real closed field, the reduction to the pure ring language needs an additional uniformity argument that is not spelled out.

    Authors: The bi-interpretation in §3 is carried out entirely in the pure ring language, without parameters or constants for a real closed field; model completeness therefore transfers directly. We will add a short paragraph after the bi-interpretation theorem clarifying this uniformity and confirming that the argument for Theorem 5.4 requires no extra reduction step. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct axiomatization of the ring-language theories of quaternions and octonions, characterizes their models via bi-interpretation with the theory of real closed fields, and derives model-theoretic properties (completeness, model completeness) from that interpretation. No equations, fitted parameters, predictions, or self-citations appear as load-bearing steps; the central claims rest on explicit definitions in the pure ring language and standard transfer results from real closed fields, with no reduction of any result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard first-order properties of real closed fields and the model theory of rings; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Real closed fields satisfy the usual first-order axioms for ordered fields with the intermediate-value property for polynomials.
    Invoked when characterizing models of the quaternion and octonion theories.

pith-pipeline@v0.9.0 · 5676 in / 1194 out tokens · 23393 ms · 2026-05-24T02:16:01.376902+00:00 · methodology

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