Studien over Topologische Algebra
David van Dantzig

Abstract:
The importance that the number continuum has for all of mathematics is mainly based on two properties. On the one hand, simple algebraic (or arithmetic) relations exist between the elements of the number continuum, i.e. relations between finitely many numbers: the real and complex numbers form a field. On the other hand, however, there exist simple topological (continuity) relations, i.e. are relations between infinitely many numbers: the real and complex numbers form a continuum. And the algebraic relations culminate in d'Alembert's theorem (the main theorem of algebra): "Every algebraic equation with complex coefficients has at least one root in the field of complex numbers." and the topological relations in the Bolzano-Weierstrass theorem: “Every bounded infinite set of numbers has at least one point of compression.” Purely algebraically, d'Alembert's theorem says that the field of complex numbers is algebraically closed; purely topologically, the Bolzano-Weierstrass theorem says that the set of real resp. complex numbers is microcompact.
In older mathematics these two aspects of the number continuum were not distinguished from each other, as e.g. is evident from the purely algebraic theorem of d'Alembert, whose proof is essentially based on the purely topological theorem of Bolzano-Weierstrass. The cause lies in the fact that the real or complex numbers themselves cannot be defined purely algebraically (based on the natural numbers), but that the convergence of certain rows of rational numbers, i.e. a topological property, must be considered.
The separation of the two aspects of the number continuum took place gradually under the influence of the axiomatization tendency of the last century; in particular it was Dedekind who, in a sense, investigated both the topological and the algebraic properties of numbers separately. Since then, topology and algebra have developed into separate branches of mathematics.
However, this creates a new problem. The number continuum has no property that distinguishes it algebraically from other algebraically closed bodies of the characteristic zero and infinite degree of transcendence. Topologically considered, there exists a multitude of geometric point sets, which have properties no less simple than this one- or two-dimensional continuum. But what is the actual cause of the fundamental role that the number continuum plays in all of mathematics?
Apparently this must be a combination of topological and algebraic properties. However, this gives rise to a new part of mathematics, topological algebra: in a set both topological and algebraic relations may be given, between which simple continuation third relations exist; one is asked to investigate and classify these (axiomatically defined) sets in general.
This research is important in another respect. In addition to the field of real and complex numbers and more or less equivalent to them, a series of other fields appear in modern number theory: those of Hensel's p-adic and p-adic numbers, which, like the field of real numbers, can be metrized by metrization. or "Bewertung" and a certain closure (completion) arise from the field of rational numbers. For the description of these fields as metric (bewertete) fields one needs the field of real numbers as an aid. In order to build up these fields without using the real numbers and to characterize them by internal properties, one must replace the metric by a topological environmental system: one thus returns to the field of topological algebra.
This dissertation will provide a brief overview of the results achieved so far. For the sake of brevity, the most important properties, known from both topology and algebra, are assumed, and most proofs are only briefly indicated. In a series of articles, which I hope to publish shortly under the title "Sur Topological Algebra," the proofs will be fully presented and the existing literature will also be referred to more fully.
In Chapter I, the concepts of T-group, T-ring and T-body, as well as the fundamental concept of completion, are introduced. Chapter II contains the theory of b_v-adic rings, which includes, among other things, the theory of Hensel's integer p-adic and p-adic numbers and Prüfer's integer "ideal" numbers as special cases. Chapter III contains some theorems about topological groups, in particular Cantorian groups. Finally, in Chapter IV the problem posed at the outset is completely resolved. And it turns out, if one disregards the trivial case that all points are isolated from each other (that there are actually no topological relations at all), that the field of complex numbers is defined by the two above-mentioned existence postulates of d'Alembert and Bolzano. Weierstrass has been fully characterized: the field of complex numbers 1s is the only mikvoperfect algebraically closed field. The positions of the fields of p-adic and p-adic numbers among the T-bodies are also indicated.

The actual reason for this study of topological algebra was the discovery that, in addition to the number continuum and the multidimensional manifolds derived from it, there are other sets, in particular the Cantorian set and the "solenoids" and "solenoidal manifolds" derived from it, which possess very simple properties, both topologically and algebraically. Shortly before, the theory of limes groups had been formulated by O. Schreier. With the help of the completion theory that we had developed at the time, B. L. van der Waerden and I tried (unsuccessfully at the time) to classify the perfectisable bodies in 1926. Subsequently, in 1928, the theory of b_v-adic rings was developed, and in the spring of 1930 the classification of microperfect bodies was achieved. The theorems about T-groups mainly date from January 1931. In the meantime, important topological-algebraic research has been published from other sides (Krull, Baer et al.).