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arxiv: 2301.00743 · v4 · submitted 2023-01-02 · 💻 cs.SC · math.RA

Computing square roots in quaternion algebras

Przemys{\l}aw Koprowski This is my paper

Pith reviewed 2026-05-24 10:11 UTC · model grok-4.3

classification 💻 cs.SC math.RA
keywords quaternion algebrassquare rootsglobal fieldsalgorithmic methodsymbolic computationlocal-global principle
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The pith

An explicit algorithm computes square roots in quaternion algebras over global fields of characteristic different from 2.

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

The paper introduces an explicit algorithmic method to compute square roots inside quaternion algebras. These algebras sit over global fields such as number fields or function fields, provided the characteristic is not 2. The method turns an otherwise non-constructive existence question into a finite sequence of steps that a computer can carry out. A reader working with algebraic computations would care because the procedure supplies a concrete way to extract roots where only theoretical guarantees existed before.

Core claim

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

What carries the argument

Algorithmic reductions combined with local-global arguments that reduce the global square-root problem to finitely many local checks inside the quaternion algebra.

If this is right

  • Square-root extraction becomes a deterministic, finite procedure rather than an existence statement.
  • The same reductions apply uniformly to both number-field and function-field cases.
  • Once a square root is found, further arithmetic operations that rely on it can be performed explicitly.

Where Pith is reading between the lines

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

  • The method may supply a template for extracting roots in other central simple algebras when similar local-global principles hold.
  • Implementation in computer-algebra systems would allow systematic testing of conjectures that involve quaternion orders.

Load-bearing premise

The base field must be a global field of characteristic different from 2 so that the reductions and local-global arguments apply.

What would settle it

Run the algorithm on an explicit quaternion algebra over the rationals or over a function field over a finite field and obtain an element whose square is not the given input.

read the original abstract

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

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 / 0 minor

Summary. The manuscript claims to present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

Significance. If a correct and efficient explicit algorithm were supplied, the result would supply a useful computational primitive for arithmetic in quaternion algebras, which appear in number theory, representation theory, and applications such as cryptography. The restriction to global fields of characteristic not 2 is the natural setting in which local-global principles for quaternion algebras are known to hold.

major comments (1)
  1. [Abstract] Abstract: the central claim is that an explicit algorithmic method exists, yet the supplied text contains no derivation steps, pseudocode, correctness argument, or reduction to known algorithms, so the soundness of the claimed method cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The primary concern raised is the absence of explicit algorithmic details in the submitted text, which prevents assessment of the method's soundness. We address this directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is that an explicit algorithmic method exists, yet the supplied text contains no derivation steps, pseudocode, correctness argument, or reduction to known algorithms, so the soundness of the claimed method cannot be assessed.

    Authors: We agree with this observation. The submitted manuscript consists only of the abstract claim without the supporting algorithmic description, derivations, pseudocode, correctness arguments, or reductions. In the revised version we will expand the paper to include a complete, self-contained presentation of the algorithm with all requested elements so that soundness can be verified. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic method is self-contained

full rationale

The paper presents an explicit algorithmic method for square-root computation in quaternion algebras over global fields of char ≠ 2. No equations, fitted parameters, predictions, or self-citations appear in the abstract or stated claim that reduce the result to its own inputs by construction. The derivation relies on standard local-global principles and field reductions that are independent of the target algorithm; the central contribution is the explicit procedure itself rather than any renamed or self-referential quantity. This is the normal case of a self-contained algorithmic paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on the standard definition of quaternion algebras and the arithmetic properties of global fields; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption The base field has characteristic different from 2
    Explicitly required by the abstract for the method to apply.
  • domain assumption The base field is global
    Stated as the setting in which the algorithm operates.

pith-pipeline@v0.9.0 · 5517 in / 1185 out tokens · 24771 ms · 2026-05-24T10:11:53.552154+00:00 · methodology

discussion (0)

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