Lehrbuch der Algebra

Abstract

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.