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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 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 (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 such that 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 (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 n-dimensional vectors form an algebra in which division (except by 0) is always possible only for n=1, 2, 4, and 8. Bott and Milnor (1958) proved that the only finite-dimensional real division algebras occur for dimensions n=1, 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.

See also

Alternative Algebra, Cayley Number, Field, Group, Jordan Algebra, Lie Algebra, Nonassociative Algebra, Power Associative Algebra, Quaternion, Schur's Lemma, Unit Ring, Zero Divisor

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References

Adams, J. F. "On the Nonexistence of Elements of Hopf Invariant One." Bull. Amer. Math. Soc. 64, 279-282, 1958.Adams, J. F. "On the Non-Existence of Elements of Hopf Invariant One." Ann. of Math. 72, 20-104, 1960.Albert, A. A. (Ed.). Studies in Modern Algebra. Washington, DC: Math. Assoc. Amer., 1963.Althoen, S. C. and Kugler, L. D. "When Is R^2 a Division Algebra?" Amer. Math. Monthly 90, 625-635, 1983.Bott, R. and Milnor, J. "On the Parallelizability of the Spheres." Bull. Amer. Math. Soc. 64, 87-89, 1958.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.Hübner, M. and Petersson, H. P. "Two-Dimensional Real Division Algebras Revisited." Beiträge zur Algebra und Geometrie 45, 29-36, 2004.Hurwitz, A. "Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln." Nachr. Königl. Gesell. Wiss. Göttingen. Math.-Phys. Klasse, 309-316, 1898.Joye, M. "Introduction élémentaire à la théorie des courbes elliptiques." http://www.dice.ucl.ac.be/crypto/introductory/courbes_elliptiques.html.Kurosh, A. G. General Algebra. New York: Chelsea, pp. 221-243, 1963.Mishchenko, A. and Solovyov, Y. "Quaternions." Quantum 11, 4-7 and 18, 2000.Petro, J. "Real Division Algebras of Dimension >1 Contain C." Amer. Math. Monthly 94, 445-449, 1987.Saltman, D. D. Lectures on Division Algebras. Providence, RI: Amer. Math. Soc., 1999.

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Division Algebra

Cite this as:

Weisstein, Eric W. "Division Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DivisionAlgebra.html

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