The Roots of Commutative Algebra in Algebraic Number Theory

Abstract

To put the issues in a broader context, these three number-theoretic problems were instrumental in the emergence of algebraic number theory-one of the two main sources of the modern discipline of commutative algebra.' The other source was algebraic geometry. It was in the setting of these two subjects that many of the main concepts and results of commutative algebra evolved in the 19th century. Thus commutative algebra can be said to have been well developed before it was created. To set the scene for our story, a word about mathematics, and especially algebra, in the 19th century. The period witnessed! fundamental transformations in mathematics -in its concepts, its methods, and in mathematicians' attitude toward their subject. These included a growing insistence on rigor and abstraction; a predisposition for founding general theories rather than focusing on specific problems; an acceptance of nonconstructive existence proofs; the emergence of mathematical specialties and specialists; the rise of the view of mathematics as a free human activity, neither deriving from, nor dependent on, nor necessarily applicable to concrete setting; the rise of set-theoretic thinking; and the reemergence of (a new variant of) the axiomatic method. As for algebra, its subject-matter and methods changed beyond recognition. Algebra as the study of solvability of polynomial equations gave way to algebra as the study of abstract structures defined axiomatically. While in the past algebra was on the periphery of mathematics, it now (the 19th and early 20th centuries) became one of its central concerns. Moreover, algebra began to penetrate other mathematical fields (e.g. geometry, analysis, logic, topology, number theory) to such an extent that in the early decades of the 20th century one began to speak of the algebraization of mathematics ([3, p. 135] and [22]).