Numbers pp 181-182 | Cite as


  • M. Koecher
  • R. Remmert
Part of the Graduate Texts in Mathematics book series (GTM, volume 123)


1. Gauss in 1831 was convinced that, outside the system of complex numbers, there were no “hypercomplex” number systems in which the basic properties of complex numbers persist; however, he expressed himself in thoroughly sibylline utterances (see 4.3.6). The Uniqueness theorem for the field ℂ appears to be a convincing pointer in support of Gauss’s thesis. In the 1880’s, a friendly dispute arose between Weierstrass and Dedekind about the proper interpretation of Gauss’s words. Described in modern language, the controversy revolved around the question of characterizing all finite-dimensional, commutative and associative ℝ algebras with unit element, divisors of zero being allowed.


Number System Unit Element Division Algebra Nilpotent Element Quaternion Algebra 
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© Springer Science+Business Media New York  1991

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  • M. Koecher
  • R. Remmert

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