Complex Number - Generalizations and Related Notions

Generalizations and Related Notions

The process of extending the field R of reals to C is known as Cayley-Dickson construction. It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively. However, with increasing dimension, the algebraic properties familiar from real and complex numbers vanish: the quaternions are only a skew field, i.e. x·y ≠ y·x for two quaternions, the multiplication of octonions fails (in addition to not being commutative) to be associative: (x·y)·z ≠ x·(y·z). However, all of these are normed division algebras over R. By Hurwitz's theorem they are the only ones. The next step in the Cayley-Dickson construction, the sedenions fail to have this structure.

The Cayley-Dickson construction is closely related to the regular representation of C, thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis 1, i. This means the following: the R-linear map

for some fixed complex number w can be represented by a 2×2 matrix (once a basis has been chosen). With respect to the basis 1, i, this matrix is

i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Any matrix

has the property that its square is the negative of the identity matrix: J2 = −I. Then

is also isomorphic to the field C, and gives an alternative complex structure on R2. This is generalized by the notion of a linear complex structure.

Hypercomplex numbers also generalize R, C, H, and O. For example this notion contains the split-complex numbers, which are elements of the ring R/(x2 − 1) (as opposed to R/(x2 + 1)). In this ring, the equation a2 = 1 has four solutions.

The field R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on Q lead to the fields Q_p of p-adic numbers (for any prime number p), which are thereby analogous to R. There are no other nontrivial ways of completing Q than R and Q_p, by Ostrowski's theorem. The algebraic closure of Q_p still carry a norm, but (unlike C) are not complete with respect to it. The completion of turns out to be algebraically closed. This field is called p-adic complex numbers by analogy.

The fields R and Q_p and their finite field extensions, including C, are local fields.