Structural algebra
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- Khan Academy - Origins of algebra (Dec. 10, 2024)
At the turn of the 20th century, algebra reflected a very clear conceptual hierarchy based on a systematically elaborated arithmetic, with a theory of polynomial equations built on top of it. Finally, a well-developed set of conceptual tools, most prominently the idea of groups, offered a comprehensive means of investigating algebraic properties. Then in 1930 a textbook was published that presented a totally new image of the discipline. This was Moderne Algebra, by the Dutch mathematician Bartel van der Waerden, who since 1924 had attended lectures in Germany by Emmy Noether at Göttingen and by Emil Artin at Hamburg. Van der Waerden’s new image of the discipline inverted the conceptual hierarchy of classical algebra. Groups, fields, rings, and other related concepts became the main focus, based on the implicit realization that all of these concepts were, in fact, instances of a more general, underlying idea: the idea of an algebraic structure. Thus, the main task of algebra became the elucidation of the properties of each of these structures and of the relationships among them. Similar questions were now asked about all these concepts, and similar concepts and techniques were used where possible. The main tasks of classical algebra became ancillary. The systems of real numbers, rational numbers, and polynomials were studied as particular instances of certain algebraic structures; the properties of these systems depended on what was known about the general structures of which they were instances, rather than the other way round.
Precursors to the structural approach
Van der Waerden’s book did not contain many new results or concepts. Its innovation lay in the unified picture it presented of the discipline of algebra. Van der Waerden brought together, in a surprisingly illuminating manner, algebraic research that had taken place over the previous three decades and in doing so he combined the contributions of several leading German algebraists from the beginning of the 20th century.
Hilbert and Steinitz
Of these German mathematicians, few were more important than David Hilbert. Among his important contributions, his work in the 1890s on the theory of algebraic number fields was decisive in establishing the conceptual approach promoted by Dedekind as dominant for several decades. As the undisputed leader of mathematics at Göttingen, then the world’s premiere research institution, Hilbert’s influence propagated through the 68 doctoral dissertations he directed as well as through the many students and mathematicians who attended his lectures. To a significant extent, the structural view of algebra was the product of some of Hilbert’s innovations, yet he basically remained a representative of the classical discipline of algebra. It is likely that the kind of algebra that developed under the influence of van der Waerden’s book had no direct appeal for Hilbert.
In 1910 Ernst Steinitz published an influential article on the abstract theory of fields that was an important milestone on the road to the structural image of algebra. His work was highly structural in that he first established the simplest kinds of subfields that any field contains and established a classification system. He then investigated how properties were passed from a field to any extension of it or to any of its subfields. In this way, he was able to characterize all possible fields abstractly. To a great extent, van der Waerden extended to the whole discipline of algebra what Steinitz accomplished for the more restricted domain of fields.
Noether and Artin
The greatest influence behind the consolidation of the structural image of algebra was no doubt Noether, who became the most prominent figure in Göttingen in the 1920s. Noether synthesized the ideas of Dedekind, Hilbert, Steinitz, and others in a series of articles in which the theory of factorization of algebraic numbers and of polynomials was masterly and succinctly subsumed under a single theory of abstract rings. She also contributed important papers to the theory of hypercomplex systems (extensions, such as the quaternions, of complex numbers to higher dimensions) that followed a similar approach, further demonstrating the potential of the structural approach.
The last significant influence on van der Waerden’s structural image of algebra was by Artin, above all for the latter’s reformulation of Galois theory. Rather than speaking of the Galois group of a polynomial equation with coefficients in a particular field, Artin focused on the group of automorphisms of the coefficients’ splitting field (the smallest extension of the field such that the polynomial could be factored into linear terms). Galois theory could then be seen as the study of the interrelations between the extensions of a field and the possible subgroups of the Galois group of the original field. In this typical structural reformulation of a classical 19th-century theory of algebra, the problem of solvability of equations by radicals appeared as a particular application of an abstract general theory.
The structural approach dominates
After the late 1930s it was clear that algebra, and in particular the structural approach within it, had become one of the most dynamic areas of research in mathematics. Structural methods, results, and concepts were actively pursued by algebraists in Germany, France, the United States, Japan, and elsewhere. The structural approach was also successfully applied to redefine other mathematical disciplines. An important early example of this was the thorough reformulation of algebraic geometry in the hands of van der Waerden, André Weil in France, and the Russian-born Oscar Zariski in Italy and the United States. In particular, they used the concepts and approach developed in ring theory by Noether and her successors. Another important example was the work of the American Marshall Stone, who in the late 1930s defined Boolean algebras, bringing under a purely algebraic framework ideas stemming from logic, topology, and algebra itself.
Over the following decades, algebra textbooks appeared around the world along the lines established by van der Waerden. Prominent among these was A Survey of Modern Algebra (1941) by Saunders Mac Lane and Garret Birkhoff, a book that was fundamental for the next several generations of mathematicians in the United States. Nevertheless, it must be stressed that not all algebraists felt, at least initially, that the new direction implied by Moderne Algebra was paramount. More classically oriented research was still being carried out well beyond the 1930s. The research of Frobenius and his former student Issai Schur, who were the most outstanding representatives of the Berlin mathematical school at the beginning of the 20th century, and of Hermann Weyl, one of Hilbert’s most prominent students, merit special mention.