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Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic
 
J. L. Lehman University of Mary Washington, Fredericksburg, VA
Quadratic Number Theory
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-4704-4737-3
Product Code:  DOL/52
List Price: $65.00
MAA Member Price: $48.75
AMS Member Price: $48.75
eBook ISBN:  978-1-4704-5155-4
Product Code:  DOL/52.E
List Price: $60.00
MAA Member Price: $45.00
AMS Member Price: $45.00
Hardcover ISBN:  978-1-4704-4737-3
eBook: ISBN:  978-1-4704-5155-4
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
Quadratic Number Theory
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Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic
J. L. Lehman University of Mary Washington, Fredericksburg, VA
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-4704-4737-3
Product Code:  DOL/52
List Price: $65.00
MAA Member Price: $48.75
AMS Member Price: $48.75
eBook ISBN:  978-1-4704-5155-4
Product Code:  DOL/52.E
List Price: $60.00
MAA Member Price: $45.00
AMS Member Price: $45.00
Hardcover ISBN:  978-1-4704-4737-3
eBook ISBN:  978-1-4704-5155-4
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 Details
     
     
    Dolciani Mathematical Expositions
    Volume: 522019; 394 pp
    MSC: 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 number-like 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:

    Readership

    Undergraduate 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 522019; 394 pp
MSC: 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 number-like 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:

Readership

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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.