On The Class Number Of Abelian Number Fields
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On the Class Number of Abelian Number Fields
With this translation, the classic monograph Über die Klassenzahl abelscher Zahlkörper by Helmut Hasse is now available in English for the first time. The book addresses three main topics: class number formulas for abelian number fields; expressions of the class number of real abelian number fields by the index of the subgroup generated by cyclotomic units; and the Hasse unit index of imaginary abelian number fields, the integrality of the relative class number formula, and the class number parity. Additionally, the book includes reprints of works by Ken-ichi Yoshino and Mikihito Hirabayashi, which extend the tables of Hasse unit indices and the relative class numbers to imaginary abelian number fields with conductor up to 100. The text provides systematic and practical methods for deriving class number formulas, determining the unit index and calculating the class number of abelian number fields. A wealth of illustrative examples, together with corrections and remarks on the original work, make this translation a valuable resource for today’s students of and researchers in number theory.
Introduction To Non-abelian Class Field Theory, An: Automorphic Forms Of Weight 1 And 2-dimensional Galois Representations
Author: Toyokazu Hiramatsu
language: en
Publisher: World Scientific
Release Date: 2016-09-13
This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.
Number Fields
Author: Frans Keune
language: en
Publisher: Radboud University Press
Release Date: 2023-03-27
Number Fields is a textbook for algebraic number theory. It grew out of lecture notes of master courses taught by the author at Radboud University, the Netherlands, over a period of more than four decades. It is self-contained in the sense that it uses only mathematics of a bachelor level, including some Galois theory. Part I of the book contains topics in basic algebraic number theory as they may be presented in a beginning master course on algebraic number theory. It includes the classification of abelian number fields by groups of Dirichlet characters. Class field theory is treated in Part II: the more advanced theory of abelian extensions of number fields in general. Full proofs of its main theorems are given using a ‘classical’ approach to class field theory, which is in a sense a natural continuation of the basic theory as presented in Part I. The classification is formulated in terms of generalized Dirichlet characters. This ‘ideal-theoretic’ version of class field theory dates from the first half of the twentieth century. In this book, it is described in modern mathematical language. Another approach, the ‘idèlic version’, uses topological algebra and group cohomology and originated halfway the last century. The last two chapters provide the connection to this more advanced idèlic version of class field theory. The book focuses on the abstract theory and contains many examples and exercises. For quadratic number fields algorithms are given for their class groups and, in the real case, for the fundamental unit. New concepts are introduced at the moment it makes a real difference to have them available.