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Computing square roots in quaternion algebras

Przemysław Koprowski

TL;DR

This work delivers the first explicit algorithm for computing square roots in quaternion algebras over global fields with $\operatorname{char} K \neq 2$. It develops separate strategies for non-central and central elements, with distinctive treatments for split and non-split algebras: a direct bi-quadratic approach for non-central elements, and isotropy/norm-equation reductions for central elements, including a two-norm-equation method in the non-split case. The algorithms leverage quadratic-form isotropy, norm equations in quadratic extensions, and local-global principles (Hilbert symbols and Hasse reciprocity) to guarantee correctness and termination. Practically, these results provide concrete procedures to compute quaternionic square roots across broad classes of global fields, enabling applications in number theory and related computational settings.

Abstract

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

Computing square roots in quaternion algebras

TL;DR

This work delivers the first explicit algorithm for computing square roots in quaternion algebras over global fields with . It develops separate strategies for non-central and central elements, with distinctive treatments for split and non-split algebras: a direct bi-quadratic approach for non-central elements, and isotropy/norm-equation reductions for central elements, including a two-norm-equation method in the non-split case. The algorithms leverage quadratic-form isotropy, norm equations in quadratic extensions, and local-global principles (Hilbert symbols and Hasse reciprocity) to guarantee correctness and termination. Practically, these results provide concrete procedures to compute quaternionic square roots across broad classes of global fields, enabling applications in number theory and related computational settings.

Abstract

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.
Paper Structure (5 sections, 1 theorem, 8 equations)

This paper contains 5 sections, 1 theorem, 8 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Square roots of non-central elements
  4. Square roots of central elements. Split case
  5. Square roots of central elements. Non-split case

Key Result

Corollary 3.2

If $\begin{tikzpicture}[yscale = -3, xscale = 3] \draw [fill = black, draw = none] svg {m 0.40860333,0.3526586 q 0.0883,-0.09292 0.19977,-0.167246 0.11618,-0.07433 0.24621,-0.08827 0.0744,-0.0093 0.13473,0.0093 0.065,0.01858 0.13007997,0.04646 0.0279,0.01394 0.0418,0.0046 0.0186,-0.0093 0.0418,-0.03

Theorems & Definitions (6)

  • Remark 2.1
  • Corollary 3.2
  • Remark 3.4
  • Remark 4.1
  • Remark 4.2
  • Remark 5.1