Division algebra
Division algebra 
touches the specialties 
is a special case of 

includes as special cases 
Division algebra is a term from the mathematical branch of abstract algebra . Roughly speaking, a division algebra is a vector space in which elements can be multiplied and divided.
Definition and example
A division algebra is not necessarily an associative algebra in which the equations and always have unique solutions for every two elements . "·" Denotes the vector multiplication in algebra. This means that algebra is free of zero divisors .
Contains the division algebra an element 1, such that for all rule, so it is called a division algebra with unity.
Example of a division algebra without a unit element with the two units and , which can be multiplied by any real numbers:
Theorems about real division algebras
A finitedimensional division algebra over the real numbers always has the dimension 1, 2, 4 or 8. This was proven in 1958 using topological methods by John Milnor and Michel Kervaire .
The four real, normalized , division algebras with one are (except for isomorphism ):
 the real numbers themselves
 the complex numbers
 the quaternions
 the octaves also have octonions or Cayley numbers.
This result is known as the Hurwitz Theorem (1898). All but the octaves satisfy the associative law of multiplication.
Every real, finitedimensional and associative division algebra is isomorphic to real numbers, complex numbers or to quaternions; this is the theorem of Frobenius (1877).
Every real, finitedimensional commutative division algebra has a maximum of dimension 2 as a vector space over the real numbers (theorem of Hopf, Heinz Hopf 1940). Associativity is not required.
Topological proofs of the existence of division algebras over the real numbers
Heinz Hopf showed in 1940 that the dimension of a division algebra must be a power of 2. In 1958, Michel Kervaire and John Milnor independently showed , using Raoul Bott's periodicity theorem on homotopy groups of the unitary and orthogonal groups, that the dimensions must be 1, 2, 4 or 8 (corresponding to the real numbers, the complex numbers, the quaternions and Octonions). The latter statement could not yet be proven purely algebraically. The proof was formulated by Michael Atiyah and Friedrich Hirzebruch with the help of the Ktheory .
For this purpose, according to Hopf, one considers the multiplication of a division algebra of dimension n over the real numbers as a continuous mapping or restricted to elements of length 1 (divide by the norm of the elements, this is not equal to zero for elements not equal to zero because a division algebra is zero divisorfree) as Illustration . Hopf proved that such an odd mapping (that is ) only exists if n is a power of 2. To do this, he used the homology groups of projective space. There are other equivalent formulations for the existence of division algebras of dimension n:
 The sphere (or the projective space ) can be parallelized (i.e. for every point x there are vectors that are linearly independent of (n1) and that depend continuously on x and are perpendicular to x).
 there are vector space bundles E over with a StiefelWhitney cohomology class not equal to zero
 there is a map with an odd Hopf invariant (see Hopf link ). Frank Adams showed that such mappings only exist for n = 2,4,8.
application
 Division algebras with one element are quasibodies (not necessarily the other way around). Hence, every example of a division algebra in synthetic geometry provides an example of an affine translation plane .
See also
literature
 Ebbinghaus et al .: Numbers . Berlin: Springer, 1992, ISBN 3540556540
 Stefaan Caenepeel, A. Verschoren Rings, Hopf Algebras, and Brauer Groups , CRC Press, 1998, ISBN 0824701534
Individual evidence
 ↑ z. B. Shafarevich, Grundzüge der Algebraischen Geometrie, Vieweg 1972, p. 201. The linear mapping (analogous for right multiplication) maps D to itself and is injective, the kernel then consists only of the zero.
 ^ Hopf, A Topological Contribution to Real Algebra, Comm. Math. Helvetici, Volume 13, 1940/41, pp. 223226
 ↑ Milnor, Some consequences of a theorem of Bott, Annals of Mathematics, Volume 68, 1958, pp. 444449
 ↑ Atiyah, Hirzebruch, Bott periodicity and the parallelisability of the spheres, Proc. Cambridge Phil. Soc., Vol. 57, 1961, pp. 223226
 ↑ The presentation of the topological proofs follows Friedrich Hirzebruch, Divisionsalgebren und Topologie (Chapter 10), in Ebbinghaus u. a. Numbers, Springer, 1983
 ^ Adams, On the nonexistence of elements of Hopf invariant one, Annals of Mathematics, Volume 72, 1960, pp. 20104
 ↑ A proof with KTheory is in Atiyah, KTheory, Benjamin 1967