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Hardcover ISBN:  9781470447373 
eBook: ISBN:  9781470451554 
Product Code:  DOL/52.B 
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AMS Member Price:  $93.75 $71.25 
Hardcover ISBN:  9781470447373 
Product Code:  DOL/52 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781470451554 
Product Code:  DOL/52.E 
List Price:  $60.00 
MAA Member Price:  $45.00 
AMS Member Price:  $45.00 
Hardcover ISBN:  9781470447373 
eBook ISBN:  9781470451554 
Product Code:  DOL/52.B 
List Price:  $125.00 $95.00 
MAA Member Price:  $93.75 $71.25 
AMS Member Price:  $93.75 $71.25 

Book DetailsDolciani Mathematical ExpositionsVolume: 52; 2019; 394 ppMSC: Primary 11
Quadratic Number Theory is an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. By restricting attention to questions about squares the author achieves the dual goals of making the presentation accessible to undergraduates and reflecting the historical roots of the subject. The representation of integers by quadratic forms is emphasized throughout the text.
Lehman introduces an innovative notation for ideals of a quadratic domain that greatly facilitates computation and he uses this to particular effect. The text has an unusual focus on actual computation. This focus, and this notation, serve the author's historical purpose as well; ideals can be seen as numberlike objects, as Kummer and Dedekind conceived of them. The notation can be adapted to quadratic forms and provides insight into the connection between quadratic forms and ideals. The computation of class groups and continued fraction representations are featured—the author's notation makes these computations particularly illuminating.
Quadratic Number Theory, with its exceptionally clear prose, hundreds of exercises, and historical motivation, would make an excellent textbook for a second undergraduate course in number theory. The clarity of the exposition would also make it a terrific choice for independent reading. It will be exceptionally useful as a fruitful launching pad for undergraduate research projects in algebraic number theory.
Ancillaries:
ReadershipUndergraduate and graduate students interested in number theory and algebraic number theory.

Table of Contents

Chapters

Introduction: A brief review of elementary number theory

Quadratic domains and ideals

Gaussian integers and sums of two squares

Quadratic domains

Ideals of quadratic domains

Quadratic forms and ideals

Quadratic forms

Correspondence between forms and ideals

Positive definite quadratic forms

Class groups of negative discriminant

Representations by positive definite forms

Class groups of quadratic subdomains

Indefinite quadratic forms

Continued fractions

Class groups of positive discriminant

Representations by indefinite forms

Quadratic recursive sequences

Properties of recursive sequences

Applications of quadratic recursive sequences

Concluding remarks


Additional Material

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Quadratic Number Theory is an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. By restricting attention to questions about squares the author achieves the dual goals of making the presentation accessible to undergraduates and reflecting the historical roots of the subject. The representation of integers by quadratic forms is emphasized throughout the text.
Lehman introduces an innovative notation for ideals of a quadratic domain that greatly facilitates computation and he uses this to particular effect. The text has an unusual focus on actual computation. This focus, and this notation, serve the author's historical purpose as well; ideals can be seen as numberlike objects, as Kummer and Dedekind conceived of them. The notation can be adapted to quadratic forms and provides insight into the connection between quadratic forms and ideals. The computation of class groups and continued fraction representations are featured—the author's notation makes these computations particularly illuminating.
Quadratic Number Theory, with its exceptionally clear prose, hundreds of exercises, and historical motivation, would make an excellent textbook for a second undergraduate course in number theory. The clarity of the exposition would also make it a terrific choice for independent reading. It will be exceptionally useful as a fruitful launching pad for undergraduate research projects in algebraic number theory.
Ancillaries:
Undergraduate and graduate students interested in number theory and algebraic number theory.

Chapters

Introduction: A brief review of elementary number theory

Quadratic domains and ideals

Gaussian integers and sums of two squares

Quadratic domains

Ideals of quadratic domains

Quadratic forms and ideals

Quadratic forms

Correspondence between forms and ideals

Positive definite quadratic forms

Class groups of negative discriminant

Representations by positive definite forms

Class groups of quadratic subdomains

Indefinite quadratic forms

Continued fractions

Class groups of positive discriminant

Representations by indefinite forms

Quadratic recursive sequences

Properties of recursive sequences

Applications of quadratic recursive sequences

Concluding remarks