Heinrich Weber – författare

Weber's three-volume set on algebra was for many years the standard text on algebra. Published at the end of the nineteenth century, it helped usher group theory to a central place in twentieth century mathematics. Volume 1 covers foundational material. Volume 2 covers group theory and its applications, plus the theory of algebraic numbers. Volume 3 covers advanced topics, such as algebraic functions, elliptic functions and class field theory. This second volume, in four parts, covers groups, linear groups, applications of group theory, and algebraic numbers. The first part, on groups, includes abelian groups, the group of a cyclotomic field, cubic and biquadratic fields and the study of general groups. The second part, on linear groups, looks at groups of linear substitutions, group invariants, polyhedral groups, and congruence groups.The third part, on applications of group theory, examines metacyclic equations and Galois groups, the inflection points of third-order curves, double-tangents of fourth-order curves, the general theory of fifth-order equations, and certain topics on equations of order seven. The fourth part begins the study of algebraic numbers (which is taken up again in the third volume). The topics here include the general theory of algebraic numbers and functions on an algebraic field, class numbers, cyclotomic fields, abelian extensions, resolvants, and transcendental numbers. Although notations have changed somewhat and algebra has become more abstract that it was in Weber's day, many of the same themes and ideas important today are central topics in Weber's book, which may be why it has become a classic.

This book is the first English translation of the classic long paper Theorie der algebraischen Functionen einer Veranderlichen (Theory of algebraic functions of one variable), published by Dedekind and Weber in 1882. The translation has been enriched by a Translator's Introduction that provides historical background, and extensive commentary embedded in the translation itself. The translation, introduction, and commentary provide the first easy access to this important paper for a wide mathematical audience: students, historians of mathematics, and professional mathematicians. Why is the Dedekind-Weber paper important? In the 1850s, Riemann initiated a revolution in algebraic geometry by interpreting algebraic curves as surfaces covering the sphere. He obtained deep and striking results in pure algebra by intuitive arguments about surfaces and their topology. However, Riemann's arguments were not rigorous, and they remained in limbo until 1882, when Dedekind and Weber put them on a sound foundation. The key to this breakthrough was to develop the theory of algebraic functions in analogy with Dedekind's theory of algebraic numbers, where the concept of ideal plays a central role. By introducing such concepts into the theory of algebraic curves, Dedekind and Weber paved the way for modern algebraic geometry.