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$\DeclareMathOperator\Mat{Mat}$How to construct an explicit isomorphism of the split quaternion algebra $(a,b)_F$ over the field $F$ to $\Mat_2(F)$?

As it is known that the algebra of quaternions is either a division ring or isomorphic to $\Mat_2(F)$ and in the second situation this quaternions are called split, I am interested in finding an algorithm to construct this explicit isomorphism.

I saw some result for $F = \mathbb{Q}(\sqrt{d})$ by Kutas, but what is the case with other fields, is there any not necessarily polynomial algorithm for them?

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  • $\begingroup$ Please be a little more specific. First, do you mean the paper eprints.sztaki.hu/9729/2/Kutas_73_30340642_ny.pdf ? Second, is $(a,b)_F = H_K(a,b)$ ? $\endgroup$
    – Dieter Kadelka
    Commented May 22 at 17:53
  • $\begingroup$ @DieterKadelka Yes on the both questions $\endgroup$
    – Samuil Lee
    Commented May 22 at 17:55
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    $\begingroup$ We know a polynomial-time algorithm over $F$ a finite field, over $F=k(t)$ where $k$ is a finite field (and maybe quadratic extensions thereof, I would need to check) for $F$ the rationals or a quadratic field, and an algorithm which is heuristically subexponential for other number fields. Note that in the rationals and quadratic cases, you are supposed to have first suceeded in proving isomorphism before trying to compute an isomorphism, which may require factorisation of some integers (also subexponential time in the worst case). $\endgroup$
    – Aurel
    Commented May 22 at 20:13
  • $\begingroup$ @Aurel Could you please share the proof of this algorithm in the case of rational numbers or maybe articles where it can be seen? Because all the notes that I have seen on this problem are quite unclear and messy $\endgroup$
    – Samuil Lee
    Commented May 23 at 15:17

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