Concrete and Abstract: Numbers, Polynomials, Rings

One obvious, main difference betweenyDmdekinkle’NwamiltiNaittlyet;getheitatimatafes ideals lies in the fact that, where, Itikhfoffri & fiNitailits properties of numbers belonging to sub-fields of the field of complex numbers, the latter is framed on an abstract, axiomatic theory built around the concept of rings. Many of the concepts introduced by Dedekind in his work on ideals, such as fields, modules,Ordnung, etc., are so closely related—from a formal point of view—to the concept of an abstract ring, that in retrospect one could easily be led to think of the process leading from research using the former concepts to the introduction of the latter as a very straightforward, almost natural step. But when one examines the actual historical process that brought about the formulation and early research on abstract rings, a completely different picture emerges. One sees that this process was rather slow and convoluted. The present chapter discusses Abraham Fraenkel’s definition of and early research of abstract rings, and the complex development of ideas immediately preceding it. It also considers some parallel advances in the theory of polynomial forms that completed the necessary background for Emmy Noether’s research on abstract rings. As will be seen, the development of the main algebraic disciplines between Dedekind and Noether involved and necessitated not only the addition of new concepts, theorems and proofs: it also implied significant qualitative changes, which modified the images of knowledge. As already stated, a central figure in this process of modification of the images of knowledge—in mathematics in general, and in algebra in particular—at the turn of the century was David Hilbert. His contribution to these developments was discussed in some detail in the preceding chapter. The present chapter opens with a discussion of Kurt Hensel’s theory of p-adic numbers and then turns to examine the way in which the theory was reformulated by his student, Abraham Fraenkel. Fraenkel tended to adopt many of the newly introduced images of algebra, which were absent from the work of his teacher.