Z, Q, R, C, H (Hamilton's quaternions), and the Cayley Numbers are the only <blank>s. What is <blank>?

Question

I know that is a special kind of field, but I don't know what it is or how it is defined.

Oh: and I might be missing another of the sets, one which contains the Cayley Numbers.

Answer 1

I don't know the exact answer and I'm pretty ignorant of algebra, but quaternions and octonions (Caley numbers) obviously aren't fields, as H is non-commutative and O is even non-associative. Further extensions of R (e.g. sedenions) behave even worse: they may be non-alternative, with zero divisors, etc.

Also, Galois fields obviously have nothing to do with this as they are _finite_. Galois extensions also have nothing to do with this as C is algebraically closed already.

Answer 2

Galois