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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