1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 73

This algebra is neither associative nor commutative. Yet it has the remark-

able property that right-and left-hand division, except by zero, are always

possible and unique.

Bott, Milnor [17], and others have shown that the only possible divi-

sion algebras over the reals, without assuming either the commutative or

associative laws of multiplication, are the reals, the complex numbers, the

quaternions, and the Cayley numbers.

A related question concerning the values of n for which the law of the

moduli

( ai + · · · + a;)(bi + · · · + b;) = ci + · · · + c;

holds was decided by A. Hurwitz in 1898. In [14] he proved that n = 1,

2, 4, and 8 are the only possible cases.

For further information about hypercomplex numbers, see (5], (8], and

(22].

1.20 AXIOMATIC FOUNDATION OF THE COMPLEX

NUMBER SYSTEM

D. H. Potts (20] and W. Bosch and P. Krajkiewicz (2] have given sets of

axioms that do not presuppose the existence of the real numbers, thus

providing an intrinsic characterization of the complex system. According

to Bosch and Krajkiewicz, a complex number system is any commutative

field F with the following property:

There exists a homomorphism T: F ~ F such that

1. T(z) f= z for at least one z E F.


2. T(T(z)) = z for all z E F.


3. The subset ~ = { z : z E F, T( z) = z} is a complete linearly ordered
   field.

The mapping Tis called the conjugate operator, and the elements z E F

are called complex numbers. There is at least one complex number system,

and any two such systems are isomorphic.

BIBLIOGRAPHY

1. R. Arens, Linear topological division algebras, Bull. Amer. Math. Soc., 53
   (1947), 623-630.


2. W. Bosch and P. Krajkiewicz, A categorical system of axioms for the complex
   numbers, Math. Mag., 43 (1970), 67-70.