See also Wedderburn's little theorem, which shows that any finite division ring is commutative and therefore a field. This is a pretty amazing result because ri

See also Wedderburn's little theorem, which shows that any finite division ring is commutative and therefore a field. This is a pretty amazing result because rings were created partly to study algebra in a non-commutative setting, and many of the most important rings, such as n x n real matrices with n > 1, are non-commutative. The quaternions in particular are a non-commutative division algebra, not subject to the theorem because infinite. The proof of Wedderburn's little theorem is relatively simple by the standards of professional math, but it's beyond me to even imagine ever coming up with it.