Division Algebra

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 Commutative. Explicitly, a division algebra is a set together with two Binary Operators $S(+,*)$ satisfying the following conditions:

1. Additive associativity: For all $a,b,c\in S$, $(a+b)+c = a+(b+c)$,

2. Additive commutativity: For all $a,b\in S$, $a+b = b+a$,

3. Additive identity: There exists an element $0\in S$ such that for all $a \in S$, $0+a=a+0=a$,

4. Additive inverse: For every $a \in S$ there exists an element $-a\in S$ such that $a+(-a)=(-a)+a=0$,

5. Multiplicative associativity: For all $a,b,c\in S$, $(a*b)*c = a*(b*c)$,

6. Multiplicative identity: There exists an element $1\in S$ not equal to 0 such that for all $a \in S$, $1*a=a*1=a$,

7. Multiplicative inverse: For every $a \in S$ not equal to 0, there exists $a^{-1}\in S$, $a*a^{-1}=a^{-1}*a=1$,

8. Left and right distributivity: For all $a,b,c\in S$, $a*(b+c)=(a*b)+(a*c)$ and $(b+c)*a=(b*a)+(c*a)$.

Thus a division algebra $(S,+,*)$ is a Unit Ring for which $(S-\{0\},*)$ 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. 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 (1956) proved that $n$-D vectors form an Algebra in which division (except by 0) is always possible only for $n=1$, 2, 4, and 8.

See also Cayley Number, Field, Group, Nonassociative Algebra, Quaternion, Unit Ring

References

Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL: University of Chicago Press, 1923.

Dixon, G. M. Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Dordrecht, Netherlands: Kluwer, 1994.

Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, pp. 326-329, 1975.

Hurwitz, A. ``Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln.'' Nachr. Gesell. Wiss. Göttingen, Math.-Phys. Klasse, 309-316, 1898.

Kurosh, A. G. General Algebra. New York: Chelsea, pp. 221-243, 1963.

Petro, J. ``Real Division Algebras of Dimension $>1$ contain $\Bbb{C}$.'' Amer. Math. Monthly 94, 445-449, 1987.