Division algebra = Field?

Question

Is a division algebra a field? If not, why does it differ?

It is an abelian group with multiplication and division. How not a field?

Answer 1

A division algebra does not need to be associative or commutative.

The quaternions are not commutative but are associative.

The octonions are neither commutative nor associative.

Answer 2

A division algebra is "essentially" an algebra with a division operation.

An algebra is "essentially" a vector space with a multiplication operation.

Any field is an algebra over itself (i.e. take both the scalars and vectors to be elements of the field -- the same way that any field is a vector space over itself, and then notice that multiplication of "vectors" is immediate since elements of a field can be multiplied with each other). Since division is also defined for elements of a field, any field also happens to be a division algebra.

However, in the same way that not all vector spaces are fields, not all algebras are fields, and not all division algebras are fields. Fields are just a special case.