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My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector spaces of the same dimension.

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  • $\begingroup$ How is the quaternion algebra defined exactly? I don't get your notation. $\endgroup$
    – Berci
    May 23, 2022 at 20:20
  • $\begingroup$ $i^2 = j^2 = k^2 = -1, ij = k ..., $ q = a +bi + cj + dk, where a,b,c,d $\in$ Zp $\endgroup$
    – Luidzzhi
    May 23, 2022 at 20:25
  • $\begingroup$ See this post. $\endgroup$ May 23, 2022 at 20:42
  • $\begingroup$ Yes, they are surely isomorphic as vector spaces. But it feels like it's rather claimed to be an isomorphism of algebras, including multiplication. $\endgroup$
    – Berci
    May 23, 2022 at 21:53
  • 1
    $\begingroup$ Please do not crosspost: mathoverflow.net/q/423172/6518 $\endgroup$
    – Kimball
    May 24, 2022 at 12:40

1 Answer 1

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Find $a,b\in \Bbb{F}_p$ such that $a^2+b^2=-1$ then let $i=\pmatrix{a&b\\b&-a},j= \pmatrix{0&1\\-1&0}$

so that $k=ij=\pmatrix{-b&a\\a&b}$ and indeed $k=-ji, i^2=j^2=k^2=-1$

If $p\ne 2$ then it will be 4-dimensional so it will span the whole of $M_2(\Bbb{F}_p)$

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