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Order Characterization of the Complex Field

Published online by Cambridge University Press:  20 November 2018

Lino Gutierrez Novoa
Lino Gutierrez Novoa*
Affiliation:
University of Alabama
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It is well known that the real number field can be characterized as an ordered field satisfied the “least upper bound” property.

Using the idea of n -ordered set, introduced in [3], and generalizing the notion of l.u.b. in a suitable way, it is possible to give a similar categorical definition of the complex field.

With these extended meanings, the main theorem of this paper (Theorem 7 in the text) is stated almost identically to the one for the real field. Any directly two-ordered field, in which the "supremum property" holds, is isomorphic to the complex field.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 3 , 01 September 1978 , pp. 313 - 318
Copyright
Copyright © Canadian Mathematical Society 1978

References

[1] Birkhoff, G., Lattice Theory, Rev. edition. Am. Math. Soc. Colloquium Publications. Vol. XXV (1948).Google Scholar
[2] Cohen, and Ehrlich, , The Structure of the Real Number System. A. Van Nostrand Co., Princeton, New Jersey (1963).Google Scholar
[3] Novoa, L. G., On n-ordered Sets and Order Completeness. Pacific Journal of Math., 15 (1965), 1337-1345.Google Scholar
[4] Novoa, L. G., Ten axioms for three dimensional Euclidean geometry. Proc. of the Am. Math. Soc. Vol. 19, No. 1 (1968), 146-152.Google Scholar