Algebraic Number Fields

Algebraic Number Fields
Author: Gerald J. Janusz
Publisher: American Mathematical Soc.
Total Pages: 288
Release: 1996
Genre: Mathematics
ISBN: 0821804294

This text presents the basic information about finite dimensional extension fields of the rational numbers, algebraic number fields, and the rings of algebraic integers in them. The important theorems regarding the units of the ring of integers and the class group are proved and illustrated with many examples given in detail. The completion of an algebraic number field at a valuation is discussed in detail and then used to provide economical proofs of global results. The book contains many concrete examples illustrating the computation of class groups, class numbers, and Hilbert class fields. Exercises are provided to indicate applications of the general theory.

The Theory of Algebraic Number Fields

The Theory of Algebraic Number Fields
Author: David Hilbert
Publisher: Springer Science & Business Media
Total Pages: 360
Release: 2013-03-14
Genre: Mathematics
ISBN: 3662035456

A translation of Hilberts "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher.

Number Fields

Number Fields
Author: Daniel A. Marcus
Publisher: Springer
Total Pages: 213
Release: 2018-07-05
Genre: Mathematics
ISBN: 3319902334

Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.

The Theory of Algebraic Numbers: Second Edition

The Theory of Algebraic Numbers: Second Edition
Author: Harry Pollard
Publisher: American Mathematical Soc.
Total Pages: 175
Release: 1975-12-31
Genre: Mathematics
ISBN: 1614440093

This monograph makes available, in English, the elementary parts of classical algebraic number theory. This second edition follows closely the plan and style of the first edition. The principal changes are the correction of misprints, the expansion or simplification of some arguments, and the omission of the final chapter on units in order to make way for the introduction of some two hundred problems.

A Conversational Introduction to Algebraic Number Theory

A Conversational Introduction to Algebraic Number Theory
Author: Paul Pollack
Publisher: American Mathematical Soc.
Total Pages: 329
Release: 2017-08-01
Genre: Mathematics
ISBN: 1470436531

Gauss famously referred to mathematics as the “queen of the sciences” and to number theory as the “queen of mathematics”. This book is an introduction to algebraic number theory, meaning the study of arithmetic in finite extensions of the rational number field Q . Originating in the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This book lays out basic results, including the three “fundamental theorems”: unique factorization of ideals, finiteness of the class number, and Dirichlet's unit theorem. While these theorems are by now quite classical, both the text and the exercises allude frequently to more recent developments. In addition to traversing the main highways, the book reveals some remarkable vistas by exploring scenic side roads. Several topics appear that are not present in the usual introductory texts. One example is the inclusion of an extensive discussion of the theory of elasticity, which provides a precise way of measuring the failure of unique factorization. The book is based on the author's notes from a course delivered at the University of Georgia; pains have been taken to preserve the conversational style of the original lectures.

A Brief Guide to Algebraic Number Theory

A Brief Guide to Algebraic Number Theory
Author: H. P. F. Swinnerton-Dyer
Publisher: Cambridge University Press
Total Pages: 164
Release: 2001-02-22
Genre: Mathematics
ISBN: 9780521004237

Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.

Cohomology of Number Fields

Cohomology of Number Fields
Author: Jürgen Neukirch
Publisher: Springer Science & Business Media
Total Pages: 831
Release: 2013-09-26
Genre: Mathematics
ISBN: 3540378898

This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.

Algebraic Number Theory

Algebraic Number Theory
Author: Edwin Weiss
Publisher: Courier Corporation
Total Pages: 308
Release: 2012-01-27
Genre: Mathematics
ISBN: 048615436X

Ideal either for classroom use or as exercises for mathematically minded individuals, this text introduces elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.