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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.

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    $\begingroup$ octonion is a division algebra. However, it isn't associative. $\endgroup$ Aug 29, 2022 at 9:58
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    $\begingroup$ The Octonions fail to be associative in an interesting way (they obey the Moufang laws). This page (math.ucr.edu/home/baez/octonions) from John Baez links some great material and also references the book by Conway and Smith. The Octonions are a division algebra. Naming conventions vary a bit - "skew" can be used to mean non-commutative in this context and division algebras can be commutative. Some people assume "algebras" are associative. So the key is to be precise about what you mean, and to read carefully what is assumed in any text, book or paper. $\endgroup$ Aug 29, 2022 at 10:28

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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.

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