Algebraic Number Theory

  title={Algebraic Number Theory},
  author={Serge Lang},
This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number… 

Tate’s Thesis

Tate’s thesis, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions (Princeton, 1950) first appeared in print as Chapter XV of the conference proceedings Algebraic Number Theory, edited by

An introduction to Tate's Thesis

In the early 20th century, Erich Hecke attempted to find a further generalization of the Dirichlet L-series and the Dedekind zeta function. In 1920, he [14] introduced the notion of a

The development of the principal genus theorem

Genus theory belongs to algebraic number theory and, in very broad terms, deals with the part of the ideal class group of a number field that is ‘easy to compute’. Historically, the importance of

A quick look at various zeta functions

The goal of this book is to guide the reader in a stroll through the garden of zeta functions of graphs. The subject arose in the late part of the last century modelled after zetas found in the other

Explicit class field theory for rational function fields

Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over F, (the finite field of q elements) and the action of the idèle

The class number one problem in function fields

In this dissertation I investigate the class number one problem in function fields. More precisely I give a survey of the current state of research into extensions of a rational function field over a

The Complex Numbers

The motivation for this chapter concerns the solvability of equations of degree larger than one which is only partly possible in the domain of real numbers. For example, no negative real number has a

Non-Free Projective Modules Over Algebraic Number Rings

This paper will provide an overview of the theory of Dedekind domains in order to present the result that the integral closure of a Dedekind domain in a finite extension of its fraction field is in

On congruences for the traces of powers of some matrices

In a series of recent papers, V.I. Arnold studied many questions concerning the statistics and dynamics of powers of elements in algebraic systems. In particular, on the basis of experimental data,


Zeta or L-functions are modelled on the Riemann’s zeta function originally defined by the series ζ(s) = P n≥1 n −s and then extended to the whole complex plane. The zeta function has an “Euler