On this wikipedia page, I read
The best-known examples of associative division algebras are the
finite-dimensional real ones (that is, algebras over the field R of
real numbers, which are finite-dimensional as a vector space over the
reals). The Frobenius theorem states that up to isomorphism there are
three such algebras: the reals themselves (dimension 1), the field of
complex numbers (dimension 2), and the quaternions (dimension 4).
First are associative division algebras and skew-fields the same thing?
Second I heard about something called the octonions and I was wondering why these are not considered yet another division algebra.
Best Answer
As mentioned in the comments, the Octonions are not associative and hence not an associative division algebra. The Sedenions are another such example; they have dimension 16 over the reals (also not a division algebra).
Mathematical terminology is not entirely standardised. Some use skew field, in fact, I knew that first. I no longer use it as I get impression that division ring or associative division algebra (according to the context) is preferred by most.
The reason may be that qualifier object is usually an object with an additional property but skew field is a field with a property removed. Of course, outside mathematics, exceptions are common e.g. a toy bird is not a bird. Maybe mathematicians are more fussy or it is just an arbitrary preference.