Quadratic Number Fields

Author : Franz Lemmermeyer
Publisher : Springer Nature
ISBN 13 : 3030786528
Total Pages : 348 pages
Book Rating : 4.26/5 ( download)

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Download or read book Quadratic Number Fields written by Franz Lemmermeyer and published by Springer Nature. This book was released on 2021-09-18 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Mak Trifković
Publisher : Springer Science & Business Media
ISBN 13 : 1461477174
Total Pages : 206 pages
Book Rating : 4.74/5 ( download)

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Download or read book Algebraic Theory of Quadratic Numbers written by Mak Trifković and published by Springer Science & Business Media. This book was released on 2013-09-14 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Tsit-Yuen Lam
Publisher : Addison-Wesley
ISBN 13 : 9780805356663
Total Pages : 344 pages
Book Rating : 4.65/5 ( download)

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Download or read book The Algebraic Theory of Quadratic Forms written by Tsit-Yuen Lam and published by Addison-Wesley. This book was released on 1980 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : J. W. S. Cassels
Publisher : Courier Dover Publications
ISBN 13 : 0486466701
Total Pages : 429 pages
Book Rating : 4.05/5 ( download)

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Download or read book Rational Quadratic Forms written by J. W. S. Cassels and published by Courier Dover Publications. This book was released on 2008-08-08 with total page 429 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : J. L. Lehman
Publisher : American Mathematical Soc.
ISBN 13 : 1470447371
Total Pages : 394 pages
Book Rating : 4.73/5 ( download)

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Download or read book Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic written by J. L. Lehman and published by American Mathematical Soc.. This book was released on 2019-02-13 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Hatice Boylan
Publisher : Springer
ISBN 13 : 3319129163
Total Pages : 150 pages
Book Rating : 4.67/5 ( download)

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Download or read book Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields written by Hatice Boylan and published by Springer. This book was released on 2014-12-05 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : David Hilbert
Publisher : Springer Science & Business Media
ISBN 13 : 3662035456
Total Pages : 360 pages
Book Rating : 4.50/5 ( download)

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Download or read book The Theory of Algebraic Number Fields written by David Hilbert and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Daniel A. Marcus
Publisher : Springer
ISBN 13 : 3319902334
Total Pages : 203 pages
Book Rating : 4.33/5 ( download)

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Download or read book Number Fields written by Daniel A. Marcus and published by Springer. This book was released on 2018-07-05 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Gerald J. Janusz
Publisher : American Mathematical Soc.
ISBN 13 : 0821804294
Total Pages : 288 pages
Book Rating : 4.92/5 ( download)

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Download or read book Algebraic Number Fields written by Gerald J. Janusz and published by American Mathematical Soc.. This book was released on 1996 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Johannes Buchmann
Publisher : Springer Science & Business Media
ISBN 13 : 3540463682
Total Pages : 328 pages
Book Rating : 4.89/5 ( download)

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Download or read book Binary Quadratic Forms written by Johannes Buchmann and published by Springer Science & Business Media. This book was released on 2007-06-22 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.