Hardcover ISBN:  9780821840757 
Product Code:  CHEL/358.H 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $49.50 
Electronic ISBN:  9781470430344 
Product Code:  CHEL/358.H.E 
List Price:  $51.00 
MAA Member Price:  $45.90 
AMS Member Price:  $45.90 

Book DetailsAMS Chelsea PublishingVolume: 358; 1967; 349 ppMSC: Primary 11;
Famous Norwegian mathematician Niels Henrik Abel advised that one should “learn from the masters, not from the pupils”. When the subject is algebraic numbers and algebraic functions, there is no greater master than Emil Artin. In this classic text, originated from the notes of the course given at Princeton University in 1950–1951 and first published in 1967, one has a beautiful introduction to the subject accompanied by Artin's unique insights and perspectives. The exposition starts with the general theory of valuation fields in Part I, proceeds to the local class field theory in Part II, and then to the theory of function fields in one variable (including the Riemann–Roch theorem and its applications) in Part III.
Prerequisites for reading the book are a standard firstyear graduate course in algebra (including some Galois theory) and elementary notions of point set topology. With many examples, this book can be used by graduate students and all mathematicians learning number theory and related areas of algebraic geometry of curves.ReadershipGraduate students and research mathematicians interested in number theory and algebraic geometry.

Table of Contents

General valuation theory

Valuations of a field

Complete fields

$e, f$ and $n$

Ramification theory

The different

Local class field theory

Preparations for local class field theory

The first and second inequalities

The norm residue symbol

The existence theorem

Applications and illustrations

Product formula and function fields in one variable

Preparations for the global theory

Characterization of fields by the product formula

Differentials in $PF$fields

The RiemannRoch theorem

Constant field extensions

Applications of the RiemannRoch theorem

Differentials in function fields

Theorems on $p$groups and Sylow groups


Additional Material

Reviews

The exposition is (as usual with Artin) quite elegant, and the parallel treatment of number fields and function can be illuminating as well as efficient ... a master of the subject ... It is a true classic in the field.
MAA Reviews 
Now, after another forty years, and being out of print for the last decades, Artin's classic of timeless beauty has been made available again for new generations of students, teachers, researchers, mathematics historians, and bibliophiles, very much to the benefit of the mathematical community as a whole.
Zentralblatt MATH


RequestsReview Copy – for reviewers who would like to review an AMS bookPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
Famous Norwegian mathematician Niels Henrik Abel advised that one should “learn from the masters, not from the pupils”. When the subject is algebraic numbers and algebraic functions, there is no greater master than Emil Artin. In this classic text, originated from the notes of the course given at Princeton University in 1950–1951 and first published in 1967, one has a beautiful introduction to the subject accompanied by Artin's unique insights and perspectives. The exposition starts with the general theory of valuation fields in Part I, proceeds to the local class field theory in Part II, and then to the theory of function fields in one variable (including the Riemann–Roch theorem and its applications) in Part III.
Prerequisites for reading the book are a standard firstyear graduate course in algebra (including some Galois theory) and elementary notions of point set topology. With many examples, this book can be used by graduate students and all mathematicians learning number theory and related areas of algebraic geometry of curves.
Graduate students and research mathematicians interested in number theory and algebraic geometry.

General valuation theory

Valuations of a field

Complete fields

$e, f$ and $n$

Ramification theory

The different

Local class field theory

Preparations for local class field theory

The first and second inequalities

The norm residue symbol

The existence theorem

Applications and illustrations

Product formula and function fields in one variable

Preparations for the global theory

Characterization of fields by the product formula

Differentials in $PF$fields

The RiemannRoch theorem

Constant field extensions

Applications of the RiemannRoch theorem

Differentials in function fields

Theorems on $p$groups and Sylow groups

The exposition is (as usual with Artin) quite elegant, and the parallel treatment of number fields and function can be illuminating as well as efficient ... a master of the subject ... It is a true classic in the field.
MAA Reviews 
Now, after another forty years, and being out of print for the last decades, Artin's classic of timeless beauty has been made available again for new generations of students, teachers, researchers, mathematics historians, and bibliophiles, very much to the benefit of the mathematical community as a whole.
Zentralblatt MATH