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.
Asked
Active
Viewed 162 times
-1
-
How is the quaternion algebra defined exactly? I don't get your notation. – Berci May 23 '22 at 20:20
-
$i^2 = j^2 = k^2 = -1, ij = k ..., $ q = a +bi + cj + dk, where a,b,c,d $\in$ Zp – Luidzzhi May 23 '22 at 20:25
-
See this post. – Dietrich Burde May 23 '22 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. – Berci May 23 '22 at 21:53
-
1Please do not crosspost: https://mathoverflow.net/q/423172/6518 – Kimball May 24 '22 at 12:40
1 Answers
2
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 $M_2(\Bbb{F}_p)$
reuns
- 77,999