Softcover ISBN:  9781470436537 
Product Code:  STML/84 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470441258 
Product Code:  STML/84.E 
List Price:  $49.00 
Individual Price:  $39.20 
Softcover ISBN:  9781470436537 
eBook: ISBN:  9781470441258 
Product Code:  STML/84.B 
List Price:  $108.00 $83.50 
Softcover ISBN:  9781470436537 
Product Code:  STML/84 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470441258 
Product Code:  STML/84.E 
List Price:  $49.00 
Individual Price:  $39.20 
Softcover ISBN:  9781470436537 
eBook ISBN:  9781470441258 
Product Code:  STML/84.B 
List Price:  $108.00 $83.50 

Book DetailsStudent Mathematical LibraryVolume: 84; 2017; 312 ppMSC: Primary 11
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 \(\mathbb{Q}\). 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.
ReadershipUndergraduate and graduate students interested in algebraic number theory.

Table of Contents

Chapters

Getting our feet wet

Cast of characters

Quadratic number fields: First steps

Paradise lost — and found

Euclidean quadratic fields

Ideal theory for quadratic fields

Prime ideals in quadratic number rings

Units in quadratic number rings

A touch of class

Measuring the failure of unique factorization

Euler’s primeproducing polynomial and the criterion of Frobenius–Rabinowitsch

Interlude: Lattice points

Back to basics: Starting over with arbitrary number fields

Integral bases: From theory to practice, and back

Ideal theory in general number rings

Finiteness of the class group and the arithmetic of $\overline {\mathbb {Z}}$

Prime decomposition in general number rings

Dirichlet’s unit theorem, I

A case study: Units in $\mathbb {Z}[\sqrt [3]{2}]$ and the Diophantine equation $X^32Y^3=\pm 1$

Dirichlet’s unit theorem, II

More Minkowski magic, with a cameo appearance by Hermite

Dedekind’s discriminant theorem

The quadratic Gauss sum

Ideal density in quadratic number fields

Dirichlet’s class number formula

Three miraculous appearances of quadratic class numbers


Additional Material

Reviews

This is a lucid, clearly written text, with a thoughtful choice and arrangement of topics, presented with contagious enthusiasm. It is a welcome addition to the existing literature on the subject.
Charles Helou, Mathematical Reviews


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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 \(\mathbb{Q}\). 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.
Undergraduate and graduate students interested in algebraic number theory.

Chapters

Getting our feet wet

Cast of characters

Quadratic number fields: First steps

Paradise lost — and found

Euclidean quadratic fields

Ideal theory for quadratic fields

Prime ideals in quadratic number rings

Units in quadratic number rings

A touch of class

Measuring the failure of unique factorization

Euler’s primeproducing polynomial and the criterion of Frobenius–Rabinowitsch

Interlude: Lattice points

Back to basics: Starting over with arbitrary number fields

Integral bases: From theory to practice, and back

Ideal theory in general number rings

Finiteness of the class group and the arithmetic of $\overline {\mathbb {Z}}$

Prime decomposition in general number rings

Dirichlet’s unit theorem, I

A case study: Units in $\mathbb {Z}[\sqrt [3]{2}]$ and the Diophantine equation $X^32Y^3=\pm 1$

Dirichlet’s unit theorem, II

More Minkowski magic, with a cameo appearance by Hermite

Dedekind’s discriminant theorem

The quadratic Gauss sum

Ideal density in quadratic number fields

Dirichlet’s class number formula

Three miraculous appearances of quadratic class numbers

This is a lucid, clearly written text, with a thoughtful choice and arrangement of topics, presented with contagious enthusiasm. It is a welcome addition to the existing literature on the subject.
Charles Helou, Mathematical Reviews