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arxiv: 1906.11076 · v1 · pith:BQKVKSXInew · submitted 2019-06-25 · 🧮 math.NT

On quaternion algebras that split over quadratic number fields

Vincenzo Acciaro , Diana Savin , Mohammed Taous , Abdelkader Zekhnini This is my paper

Pith reviewed 2026-05-25 16:38 UTC · model grok-4.3

classification 🧮 math.NT
keywords quaternion algebrasquadratic number fieldssplitting conditionsdivision algebrasHilbert symbolramificationFibonacci primescentral simple algebras
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The pith

Necessary and sufficient conditions determine when the quaternion algebra H_K(α, m) splits over the quadratic field K.

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

The paper establishes necessary and sufficient conditions for the quaternion algebra H_K(α, m) over K = Q(√d) to split, along with a complete characterization of when it is a division algebra if α and m are distinct positive primes. These conditions rely on the field's ramification behavior and explicit norm computations. A sympathetic reader cares because the result classifies central simple algebras over quadratic fields in concrete cases, including examples built from prime Fibonacci numbers. The work extends the standard theory of quaternion algebras by handling specific parameter choices.

Core claim

We give necessary and sufficient conditions for H_K(α, m) to split over K for some values of α, and we obtain a complete characterization of division quaternion algebras H_K(α, m) over K whenever α and m are two distinct positive prime integers.

What carries the argument

The quaternion algebra H_K(α, m) over the ring of integers of K, whose splitting is decided by the Hilbert symbol and ramification at primes of K.

If this is right

  • For any two distinct positive primes α and m one can decide whether H_K(α, m) is a division algebra or isomorphic to the matrix algebra over K.
  • The same criteria apply to certain non-prime square-free α when additional local conditions at the primes dividing d hold.
  • Concrete examples arise when α or m is a prime Fibonacci number, yielding explicit split or division cases.
  • The Brauer class of H_K(α, m) is trivial precisely when the listed local invariants vanish.

Where Pith is reading between the lines

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

  • The same symbol computations could be reused to test splitting for parameters that are products of a few small primes.
  • The characterization supplies an explicit way to list all division quaternion algebras over a given quadratic field with prime parameters.
  • One could check whether the same conditions remain valid when K is replaced by a cubic extension.

Load-bearing premise

The explicit norm-residue or Hilbert-symbol calculations over K correctly capture the ramification at all relevant primes.

What would settle it

An explicit pair of distinct positive primes α and m together with a quadratic field K where the paper's predicted splitting behavior disagrees with direct computation of the Hilbert symbol at one prime of K.

read the original abstract

Let $d$ and $m$ be two distinct squarefree integers and $\mathcal{O}_K$ the ring of integers of the quadratic field $K=\mathbb{Q}(\sqrt{d})$. Denote by $ H_K(\alpha, m)$ a quaternion algebra over $K$, where $\alpha\in \mathcal{O}_K$. In this paper we give necessary and sufficient conditions for $ H_K(\alpha, m)$ to split over $K$ for some values of $\alpha$, and we obtain a complete characterization of division quaternion algebras $ H_K(\alpha, m)$ over $K$ whenever $\alpha$ and $m$ are two distinct positive prime integers. Examples are given involving prime Fibonacci numbers.

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

1 major / 1 minor

Summary. The manuscript claims to provide necessary and sufficient conditions for the quaternion algebra H_K(α, m) over K = Q(√d) to split for some values of α, together with a complete characterization of the cases in which H_K(α, m) is a division algebra when α and m are distinct positive prime integers; examples involving prime Fibonacci numbers are included.

Significance. If the explicit local computations are correct, the results would supply concrete, usable criteria for ramification of quaternion algebras over quadratic fields, extending standard Hilbert-symbol techniques to this setting and potentially aiding work on central simple algebras and class field theory.

major comments (1)
  1. [Proofs of the main theorems (characterization for prime α, m)] The complete characterization of division algebras for prime α, m rests on explicit evaluations of the Hilbert symbol (α, m)_v at all places of K (including those dividing 2dαm). The manuscript must verify each local computation; an undetected arithmetic error in even one such evaluation would falsify the claimed necessary-and-sufficient conditions for the affected pairs.
minor comments (1)
  1. [Abstract] The notation H_K(α, m) is introduced without an explicit definition in the abstract; the reader must infer it from standard quaternion-algebra conventions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and for highlighting the importance of verifying the local Hilbert-symbol computations that underpin the main characterization theorems. We address this point directly below.

read point-by-point responses

  1. Referee: [Proofs of the main theorems (characterization for prime α, m)] The complete characterization of division algebras for prime α, m rests on explicit evaluations of the Hilbert symbol (α, m)_v at all places of K (including those dividing 2dαm). The manuscript must verify each local computation; an undetected arithmetic error in even one such evaluation would falsify the claimed necessary-and-sufficient conditions for the affected pairs.

    Authors: The proofs of the main results (Theorems 4.1 and 5.3) proceed by exhaustive case analysis of the Hilbert symbol (α, m)_v at every place v of K. For each place we record the explicit local computation, invoking the standard formulas for the Hilbert symbol over Q_p, the behavior under unramified and ramified quadratic extensions, and the explicit description of the primes dividing 2dαm. These calculations occupy the bulk of Sections 4 and 5 and are presented in sufficient detail that each step (valuation, residue-field computation, and application of the product formula) can be checked independently. We have re-verified all of them during manuscript preparation. If the referee can point to a particular place or pair (α, m) where an error is suspected, we will supply any omitted intermediate arithmetic or correct the statement accordingly. revision: no

Circularity Check

0 steps flagged

No circularity; direct derivation from standard algebraic number theory

full rationale

The paper derives necessary and sufficient conditions for quaternion algebras H_K(α, m) to split over quadratic fields K=Q(√d) using ramification theory, norm-residue symbols, and Hilbert symbols at places of K. No equations or claims reduce results to fitted parameters, self-definitions, or load-bearing self-citations; the characterization for prime α, m rests on explicit but standard local computations from field theory. The derivation is self-contained against external benchmarks in algebraic number theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard construction of quaternion algebras via Hilbert symbols and the ramification theory of quadratic extensions; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard properties of the Hilbert symbol and ramification in quadratic extensions of Q
    Invoked implicitly in the definition of H_K(α, m) and the splitting criterion.

pith-pipeline@v0.9.0 · 5650 in / 1175 out tokens · 25430 ms · 2026-05-25T16:38:01.555644+00:00 · methodology

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Reference graph

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