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.
2026-04-06 06:31:42.1775457102
How to prove that Quaternion's algebra over isomorphic to Mat2(Z
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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 $M_2(\Bbb{F}_p)$