Quadratic Number Fields
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Author |
: Franz Lemmermeyer |
Publisher |
: Springer Nature |
Total Pages |
: 348 |
Release |
: 2021-09-18 |
ISBN-10 |
: 9783030786526 |
ISBN-13 |
: 3030786528 |
Rating |
: 4/5 (26 Downloads) |
Synopsis Quadratic Number Fields by : Franz Lemmermeyer
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 |
Total Pages |
: 206 |
Release |
: 2013-09-14 |
ISBN-10 |
: 9781461477174 |
ISBN-13 |
: 1461477174 |
Rating |
: 4/5 (74 Downloads) |
Synopsis Algebraic Theory of Quadratic Numbers by : Mak Trifković
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 |
: Johannes Buchmann |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 328 |
Release |
: 2007-06-22 |
ISBN-10 |
: 9783540463689 |
ISBN-13 |
: 3540463682 |
Rating |
: 4/5 (89 Downloads) |
Synopsis Binary Quadratic Forms by : Johannes Buchmann
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.
Author |
: J. L. Lehman |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 410 |
Release |
: 2019-02-13 |
ISBN-10 |
: 9781470447373 |
ISBN-13 |
: 1470447371 |
Rating |
: 4/5 (73 Downloads) |
Synopsis Quadratic Number Theory by : J. L. Lehman
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 |
: Tsit-Yuen Lam |
Publisher |
: Addison-Wesley |
Total Pages |
: 344 |
Release |
: 1980 |
ISBN-10 |
: 0805356665 |
ISBN-13 |
: 9780805356663 |
Rating |
: 4/5 (65 Downloads) |
Synopsis The Algebraic Theory of Quadratic Forms by : Tsit-Yuen Lam
Author |
: J. W. S. Cassels |
Publisher |
: Courier Dover Publications |
Total Pages |
: 429 |
Release |
: 2008-08-08 |
ISBN-10 |
: 9780486466705 |
ISBN-13 |
: 0486466701 |
Rating |
: 4/5 (05 Downloads) |
Synopsis Rational Quadratic Forms by : J. W. S. Cassels
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 |
: Duncan A. Buell |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 249 |
Release |
: 2012-12-06 |
ISBN-10 |
: 9781461245421 |
ISBN-13 |
: 1461245427 |
Rating |
: 4/5 (21 Downloads) |
Synopsis Binary Quadratic Forms by : Duncan A. Buell
The first coherent exposition of the theory of binary quadratic forms was given by Gauss in the Disqnisitiones Arithmeticae. During the nine teenth century, as the theory of ideals and the rudiments of algebraic number theory were developed, it became clear that this theory of bi nary quadratic forms, so elementary and computationally explicit, was indeed just a special case of a much more elega,nt and abstract theory which, unfortunately, is not computationally explicit. In recent years the original theory has been laid aside. Gauss's proofs, which involved brute force computations that can be done in what is essentially a two dimensional vector space, have been dropped in favor of n-dimensional arguments which prove the general theorems of algebraic number the ory. In consequence, this elegant, yet pleasantly simple, theory has been neglected even as some of its results have become extremely useful in certain computations. I find this neglect unfortunate, because binary quadratic forms have two distinct attractions. First, the subject involves explicit computa tion and many of the computer programs can be quite simple. The use of computers in experimenting with examples is both meaningful and enjoyable; one can actually discover interesting results by com puting examples, noticing patterns in the "data," and then proving that the patterns result from the conclusion of some provable theorem.
Author |
: David Hilbert |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 360 |
Release |
: 2013-03-14 |
ISBN-10 |
: 9783662035450 |
ISBN-13 |
: 3662035456 |
Rating |
: 4/5 (50 Downloads) |
Synopsis The Theory of Algebraic Number Fields by : David Hilbert
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 |
: Fred Wayne Dodd |
Publisher |
: |
Total Pages |
: 168 |
Release |
: 1983 |
ISBN-10 |
: UCAL:B4178432 |
ISBN-13 |
: |
Rating |
: 4/5 (32 Downloads) |
Synopsis Number Theory in the Quadratic Field with Golden Section Unit by : Fred Wayne Dodd
Author |
: M. Ishida |
Publisher |
: Springer |
Total Pages |
: 123 |
Release |
: 2006-12-08 |
ISBN-10 |
: 9783540375531 |
ISBN-13 |
: 3540375538 |
Rating |
: 4/5 (31 Downloads) |
Synopsis The Genus Fields of Algebraic Number Fields by : M. Ishida
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