These codes have been studied more widely, and division algebras over number fields have now become the standard tool for constructing such codes.
22.
There are only three finite-dimensional associative division algebras over the reals-the real numbers, the complex numbers and the quaternions.
23.
If " k " is the field of complex numbers, the only option is that this division algebra is the complex numbers.
24.
In other words, the only complex Banach algebra that is a division algebra is the complex numbers "'C " '.
25.
While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.
26.
A seminal paper by Noether, Helmut Hasse, and Richard Brauer pertains to division algebras, which are algebraic systems in which division is possible.
27.
An immediate example of simple algebras are division algebras, where every element has a multiplicative inverse, for instance, the real algebra of quaternions.
28.
If the division algebra is not assumed to be associative, usually some weaker condition ( such as alternativity or power associativity ) is imposed instead.
29.
Later work showed that in fact, any finite-dimensional real division algebra must be of dimension 1, 2, 4, or 8.
30.
Also, we have the notable theorem of Frobenius that there are exactly three real associative division algebras : real numbers, complex numbers, and quaternions.
