**Dolciani Mathematical Expositions**

Volume: 52;
2019;
394 pp;
Hardcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-4737-3
Product Code: DOL/52**

List Price: $52.00

AMS Member Price: $39.00

MAA Member Price: $39.00

**Electronic ISBN: 978-1-4704-5155-4
Product Code: DOL/52.E**

List Price: $52.00

AMS Member Price: $39.00

MAA Member Price: $39.00

#### Supplemental Materials

# Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic

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*J. L. Lehman*

MAA Press: An Imprint of the American Mathematical Society

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.

A solutions
manual is freely available electronically.

#### Readership

Undergraduate and graduate students interested in number theory and algebraic number theory.

#### Table of Contents

# Table of Contents

## Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic

- Cover Cover11
- Title page i2
- Copyright ii3
- Contents iii4
- Preface vii8
- Introduction: A Brief Review of Elementary Number Theory 116
- Part One: Quadratic Domains and Ideals 1530
- Chapter 1. Gaussian Integers and Sums of Two Squares 1732
- 1.1. Sums of Two Squares 1732
- 1.2. Gaussian Integers 2540
- 1.3. Ideal Form for Gaussian Integers 3247
- 1.4. Factorization and Multiplication with Ideal Forms 3853
- 1.5. Reduction of Ideal Forms for Gaussian Integers 4459
- 1.6. Sums of Two Squares Revisited 4863
- Gaussian Integers and Sums of Two Squares—Review 5267

- Chapter 2. Quadratic Domains 5570
- 2.1. Quadratic Numbers and Quadratic Integers 5671
- 2.2. Domains of Quadratic Integers 6176
- 2.3. Ideal Form for Quadratic Integers 6883
- 2.4. Ideal Numbers 7489
- 2.5. Quadratic Domains with Unique Factorization 8095
- 2.6. Quadratic Domains without Unique Factorization 86101
- Quadratic Domains—Review 91106

- Chapter 3. Ideals of Quadratic Domains 93108
- Part Two: Quadratic Forms and Ideals 129144
- Chapter 4. Quadratic Forms 131146
- Chapter 5. Correspondence between Forms and Ideals 153168
- Part Three: Positive Definite Quadratic Forms 177192
- Chapter 6. Class Groups of Negative Discriminant 179194
- Chapter 7. Representations by Positive Definite Forms 199214
- Chapter 8. Class Groups of Quadratic Subdomains 227242
- Part Four: Indefinite Quadratic Forms 245260
- Chapter 9. Continued Fractions 247262
- 9.1. Introduction to Continued Fractions 247262
- 9.2. Pell’s Equation 253268
- 9.3. Convergence of Continued Fractions 259274
- 9.4. Continued Fraction Expansions of Real Numbers 265280
- 9.5. Purely Periodic Continued Fractions 269284
- 9.6. Continued Fractions of Irrational Quadratic Numbers 274289
- Continued Fractions—Review 281296

- Chapter 10. Class Groups of Positive Discriminant 285300
- Chapter 11. Representations by Indefinite Forms 307322
- Part Five: Quadratic Recursive Sequences 333348
- Chapter 12. Properties of Recursive Sequences 335350
- Chapter 13. Applications of Quadratic Recursive Sequences 367382
- Concluding Remarks 383398
- List of Notation 387402
- Index 391406
- Back Cover Back Cover1410