Quadratic Number Fields
Quadratic Number Fields

Author: Franz Lemmermeyer

Publisher: Springer Nature

Published: 2021-09-18

Total Pages: 348

ISBN-13: 3030786528

DOWNLOAD EBOOK

This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.

Algebraic Theory of Quadratic Numbers
Algebraic Theory of Quadratic Numbers

Author: Mak Trifković

Publisher: Springer Science & Business Media

Published: 2013-09-14

Total Pages: 206

ISBN-13: 1461477174

DOWNLOAD EBOOK

By focusing on quadratic numbers, this advanced undergraduate or master’s level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes. The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory.

The Theory of Algebraic Number Fields
The Theory of Algebraic Number Fields

Author: David Hilbert

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 360

ISBN-13: 3662035456

DOWNLOAD EBOOK

A translation of Hilberts "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher.

Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields
Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields

Author: Hatice Boylan

Publisher: Springer

Published: 2014-12-05

Total Pages: 150

ISBN-13: 3319129163

DOWNLOAD EBOOK

The new theory of Jacobi forms over totally real number fields introduced in this monograph is expected to give further insight into the arithmetic theory of Hilbert modular forms, its L-series, and into elliptic curves over number fields. This work is inspired by the classical theory of Jacobi forms over the rational numbers, which is an indispensable tool in the arithmetic theory of elliptic modular forms, elliptic curves, and in many other disciplines in mathematics and physics. Jacobi forms can be viewed as vector valued modular forms which take values in so-called Weil representations. Accordingly, the first two chapters develop the theory of finite quadratic modules and associated Weil representations over number fields. This part might also be interesting for those who are merely interested in the representation theory of Hilbert modular groups. One of the main applications is the complete classification of Jacobi forms of singular weight over an arbitrary totally real number field.

Binary Quadratic Forms
Binary Quadratic Forms

Author: Johannes Buchmann

Publisher: Springer Science & Business Media

Published: 2007-06-22

Total Pages: 328

ISBN-13: 3540463682

DOWNLOAD EBOOK

The book deals with algorithmic problems related to binary quadratic forms. It uniquely focuses on the algorithmic aspects of the theory. The book introduces the reader to important areas of number theory such as diophantine equations, reduction theory of quadratic forms, geometry of numbers and algebraic number theory. The book explains applications to cryptography and requires only basic mathematical knowledge. The author is a world leader in number theory.

Number Fields
Number Fields

Author: Daniel A. Marcus

Publisher: Springer

Published: 2018-07-05

Total Pages: 213

ISBN-13: 3319902334

DOWNLOAD EBOOK

Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.

Algebraic Number Theory
Algebraic Number Theory

Author: Richard A. Mollin

Publisher: CRC Press

Published: 2011-01-05

Total Pages: 424

ISBN-13: 1439845999

DOWNLOAD EBOOK

Bringing the material up to date to reflect modern applications, this second edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. It offers a more complete and involved treatment of Galois theory, a more comprehensive section on Pollard's cubic factoring algorithm, and more detailed explanations of proofs to provide a sound understanding of challenging material. This edition also studies binary quadratic forms and compares the ideal and form class groups. The text includes convenient cross-referencing, a comprehensive index, and numerous exercises and applications.

Algebraic Number Fields
Algebraic Number Fields

Author: Gerald J. Janusz

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 288

ISBN-13: 0821804294

DOWNLOAD EBOOK

This text presents the basic information about finite dimensional extension fields of the rational numbers, algebraic number fields, and the rings of algebraic integers in them. The important theorems regarding the units of the ring of integers and the class group are proved and illustrated with many examples given in detail. The completion of an algebraic number field at a valuation is discussed in detail and then used to provide economical proofs of global results. The book contains many concrete examples illustrating the computation of class groups, class numbers, and Hilbert class fields. Exercises are provided to indicate applications of the general theory.

Rational Quadratic Forms
Rational Quadratic Forms

Author: J. W. S. Cassels

Publisher: Courier Dover Publications

Published: 2008-08-08

Total Pages: 429

ISBN-13: 0486466701

DOWNLOAD EBOOK

Exploration of quadratic forms over rational numbers and rational integers offers elementary introduction. Covers quadratic forms over local fields, forms with integral coefficients, reduction theory for definite forms, more. 1968 edition.