Division Algebra
A division algebra over a field
is an associative -algebra with such that every nonzero element has a multiplicative inverse.
e.g.
- A field is simply a commutative division algebra;
is a division algebra over , with dimension 2; - The quaternions
is a 4-dimensional division algebra over .
Theorem
Every finite-dimensional division algebra over an algebraically closed field is just that field itself.
Frobenius Theorem
Every finite-dimensional division algebra over
is isomorphic to , or .