Class Field Theory. Field Extensions

Abstract

In this brief survey on class field theory and related questions we mainly present the papers reviewed in the “Mathematics” section of Referativnyi Zhurnal during 1958–1967. Among the books published during this time we note those by Chevalley [20] (a systematic exposition and application of cohomology groups), Artin and Tate [12] (the most modern exposition of class field theory with the application of cohomology groups and of class systems; the ground field is either the field of algebraic numbers of finite degree or the field of functions of one variable over a finite field of constants), and Serre [99] (the connection of class field theory with algebraic curves). In 1963 there appeared a survey by Ribenboim [93] who set forth the results in the Hilbert — Takagi theory and the reciprocity law of Artin and who examined the problem of finding functions whose values could be generated by any Abelian extension of the field of algebraic numbers. On local class field theory we have the survey by Hochschild [43] (also see Samuel’s report [94] on Hochschild’s results: G. Hochschild, “Local class field theory,” Ann. Math., 51(2):331–347 (1950), where local class field theory is presented with the help of cohomologies).