# How to prove that Quaternion's algebra over isomorphic to Mat2(Z [duplicate]

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.

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

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 $$M_2(\Bbb{F}_p)$$