Division algebra

Prejudice to the special areas

• Mathematics Abstract Algebra
• Linear Algebra

Is a special case of

• Algebra
• Quasi body ( for division algebra with identity )

Includes as special cases

• Skew field
• Octaves

Division algebra is a term from the mathematical branch of abstract algebra. Roughly speaking, it is in a division algebra is a vector space in which you can multiply and divide elements.

Definition and Example

A division algebra is a not necessarily associative algebra in which any two elements have the equations and consistent and unique solutions. It refers to "·" the vector multiplication in the algebra.

Contains the division algebra is 1, so that is true, it is called a division algebra with identity.

Example of a division algebra without identity with the two units and which can be multiplied by arbitrary real numbers:

Sentences about real division algebras

A finite-dimensional division algebra over the real numbers always has the dimension 1, 2, 4 or 8 This was demonstrated in 1958 with topological methods by John Milnor and Kervaire Michel.

The four real, normalized, with one division algebras are ( up to isomorphism ):

• The real numbers themselves
• The complex numbers
• The quaternions
• The octaves or octonions also Cayley numbers.

This result is known as a set of Hurwitz (1898 ).

Every real, finite-dimensional and associative division algebra is isomorphic to the real numbers, the complex numbers or the quaternions; this is the Frobenius theorem (1877 ).

Application

• Division algebras with identity are quasi- body (not necessarily vice versa). Therefore each example of a division in the synthetic algebra geometry provides an example of an affine translation level.