Introduction to the Construction of Class Fields

by Cohn, Harvey

• ISBN13: 9780486683461
• ISBN10: 048668346X
• Edition: Revised
• Format: Paperback
• Copyright: 1995-01-04
• Publisher: Dover Publications

Summary

A broad introduction to quadratic forms, modular functions, interpretation by rings and ideals, class fields by radicals and more. 1985 ed.

Table of Contents

Preface

Preview: Quadratic forms and modular functions Exercise

Bibliographic note

Early versions of class field theory

Rational definition of class field

Class fields for quadratic forms

Dirichlet characters

Appendix on rational number theory Exercises

Bibliographic note

Interpretation by rings and ideals

Rings and modules

Ideals and divisors

Ideals as homomorphisms Exercises

Bibliographic note

Finite invariants of a field

Integers and discriminants

Units and regulators

The zeta-function and L-Functions Exercises

Bibliographic note

Function fields

Rational functions

Riemann surfaces

Elliptic functions

Integer rings Exercises

Bibliographic note

Relative fields

Discriminant and different

Hilbert sequence of fields and groups

Classical illustrations

The Artin symbol Exercises

Bibliographic note

The WHAT theorem of class field theory

The two class field concepts

Ray class structure

The conductor

The WHAT theorem summarized

Analytic class field theory Exercises

Bibliographic note

The genus class field and transfer theory

Ring class structure and genus

Absolute Galois structures

Class transfer theory Exercises

Bibliographic note

Class fields by radicals

An unramified class field tower

A ramified tower

Hilbert's relative units

The governing field Exercises

Bibliographic note

The modular function field

Summation formulas

Special matrix and transformation groups

Modular fields and subfields

Modular equations Exercises

Bibliographic note

Class fields by modular functions

Modular invariants

Weber's theorem

Iteration and the icosahedron

Epilogue to Weber's theorem Exercises

Bibliographic note

References

Index

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