Some authors omit the associativity requirement and define a division algebra to be an algebra D over a field such that for any element a in D and any nonzero element b in D there exists precisely one element x with a = bx and precisely one element y in D such that a = yb. In the remainder of this article, we will however assume associativity.
The prototypical example of a division algebra over the real numbers is given by the quaternions. Every field extension forms a division algebra over the ground field. There is no finitedimensional division algebra over the complex numbers (except for the complex numbers themselves). The only finitedimensional division algebras over the real numbers (up to algebra isomorphism) are:
See also: normed division algebra, division, division ring
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