by

Ciro Santilli (@cirosantilli, 37)

Algebra over a field

A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

complex numbers, i.e. $R^{2}$ with complex number multiplication
$R^{3}$ with the cross product
quaternions, i.e. $R^{4}$ with the quaternion multiplication

Table of contents
- Division algebra Algebra over a field
  - Frobenius theorem (real division algebras) Division algebra

Division algebra

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An algebra over a field where division exists.

Frobenius theorem (real division algebras, Classification of associative real division algebras)

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There are 3: real numbers, complex numbers and quaternions.

Notably, the octonions are not associative.

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