Lehman J.L. Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic

Washington: MAA Press, 2019. — 409 p. — (Dolciani Mathematical Expositions, 52). — ISBN: 1470447371.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.Preface.
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.
List of Notation.
Index.True PDF