A division algebra, also called a "division ring" or "skew field," is a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative. Every field is therefore also a division algebra. In French, the term "corps non commutatif" is used to mean division algebra, while "corps" alone means field.
Explicitly, a division algebra is a set together with two binary operators satisfying the following conditions:
1. Additive associativity: For all , .
2. Additive commutativity: For all , .
3. Additive identity: There exists an element such that for all , .
4. Additive inverse: For every there exists an element such that .
5. Multiplicative associativity: For all , .
6. Multiplicative identity: There exists an element not equal to 0 such that for all , .
7. Multiplicative inverse: For every not equal to 0, there exists such that .
8. Left and right distributivity: For all , and .
Thus a division algebra is a unit ring for which is a group. A division algebra must contain at least two elements. A commutative division algebra is called a field.
In 1878 and 1880, Frobenius and Peirce proved that the only associative real division algebras are real numbers, complex numbers, and quaternions (Mishchenko and Solovyov 2000). The Cayley algebra is the only nonassociative division algebra. Hurwitz (1898) proved that the algebras of real numbers, complex numbers, quaternions, and Cayley numbers are the only ones where multiplication by unit "vectors" is distance-preserving.
Adams (1958, 1960) proved that -dimensional vectors form an algebra in which division (except by 0) is always possible only for , 2, 4, and 8. Bott and Milnor (1958) proved that the only finite-dimensional real division algebras occur for dimensions , 2, 4, and 8. Each gives rise to an algebra with particularly useful physical applications (which, however, is not itself necessarily nonassociative), and these four cases correspond to real numbers, complex numbers, quaternions, and Cayley numbers, respectively.