Abstract
We take ℝ as the basie field, though in place of ℝ one could equally well have chosen any commutative field K. Real numbers will always be denoted in Chapters 7 to 11 by small Greek letters. Every n-dimensional ℝ-vector space is isomorphic to the number space ℝn of n-tuples x = (ξ1,...,ξn).
Die größten und fruchtbarsten Fortschritte in der Mathematik sind vorzugsweise durch die Schöpfung neuer Begriffe gemacht, nachdem die häufige Wiederkehr zusammengesetzter Erscheinungen dazu gedrängt hat (R. Dedekind, Was sind und was sollen die Zahlen? 1888).
[The greatest and most fruitful advances in Mathematics are chiefly made by the creation of new ideas and concepts, after the frequent reoccurrence of composite (or complex) phenomena has driven us to this. (R. Dedekind: What are numbers, and wh at are they good for?]
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Additional Reading
I.L. Kantor and A.S. Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Springer-Verlag (1989)
K.H. Parshall: In pursuit of the finite division algebra theorem and beyond: Joseph H.M. Wedderburn, Leonard E. Dickson and Oswald Veblen, Arch. Internat. Hist. Sci. 33, no. 111 (1983), 274–299.
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Koecher, M., Remmert, R. (1991). Repertory. Basic Concepts from the Theory of Algebras. In: Numbers. Graduate Texts in Mathematics, vol 123. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1005-4_9
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