Algebraic Theory Of Quadratic Numbers PDF Download
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| Author | : Mak Trifković |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 206 |
| Release | : 2013-09-14 |
| Genre | : Mathematics |
| ISBN | : 1461477174 |
Download Algebraic Theory of Quadratic Numbers Book in PDF, ePub and Kindle
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 | : Mak Trifkovi |
| Publisher | : |
| Total Pages | : 212 |
| Release | : 2013-08-31 |
| Genre | : |
| ISBN | : 9781461477181 |
Download Algebraic Theory of Quadratic Numbers Book in PDF, ePub and Kindle
| Author | : Franz Lemmermeyer |
| Publisher | : Springer Nature |
| Total Pages | : 348 |
| Release | : 2021-09-18 |
| Genre | : Mathematics |
| ISBN | : 3030786528 |
Download Quadratic Number Fields Book in PDF, ePub and Kindle
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 | : Kazimierz Szymiczek |
| Publisher | : Routledge |
| Total Pages | : 508 |
| Release | : 2017-11-22 |
| Genre | : Mathematics |
| ISBN | : 1351464205 |
Download Bilinear Algebra Book in PDF, ePub and Kindle
Giving an easily accessible elementary introduction to the algebraic theory of quadratic forms, this book covers both Witt's theory and Pfister's theory of quadratic forms. Leading topics include the geometry of bilinear spaces, classification of bilinear spaces up to isometry depending on the ground field, formally real fields, Pfister forms, the Witt ring of an arbitrary field (characteristic two included), prime ideals of the Witt ring, Brauer group of a field, Hasse and Witt invariants of quadratic forms, and equivalence of fields with respect to quadratic forms. Problem sections are included at the end of each chapter. There are two appendices: the first gives a treatment of Hasse and Witt invariants in the language of Steinberg symbols, and the second contains some more advanced problems in 10 groups, including the u-invariant, reduced and stable Witt rings, and Witt equivalence of fields.
| Author | : J. L. Lehman |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 394 |
| Release | : 2019-02-13 |
| Genre | : Algebraic fields |
| ISBN | : 1470447371 |
Download Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic Book in PDF, ePub and Kindle
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 | : Paulo Ribenboim |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 676 |
| Release | : 2013-11-11 |
| Genre | : Mathematics |
| ISBN | : 0387216901 |
Download Classical Theory of Algebraic Numbers Book in PDF, ePub and Kindle
The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics.
| Author | : Richard S. Elman |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 456 |
| Release | : 2008-07-15 |
| Genre | : Mathematics |
| ISBN | : 9780821873229 |
Download The Algebraic and Geometric Theory of Quadratic Forms Book in PDF, ePub and Kindle
This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given. Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.
| Author | : Oleg T. Izhboldin |
| Publisher | : Springer |
| Total Pages | : 198 |
| Release | : 2004-02-07 |
| Genre | : Mathematics |
| ISBN | : 3540409904 |
Download Geometric Methods in the Algebraic Theory of Quadratic Forms Book in PDF, ePub and Kindle
The geometric approach to the algebraic theory of quadratic forms is the study of projective quadrics over arbitrary fields. Function fields of quadrics have been central to the proofs of fundamental results since the 1960's. Recently, more refined geometric tools have been brought to bear on this topic, such as Chow groups and motives, and have produced remarkable advances on a number of outstanding problems. Several aspects of these new methods are addressed in this volume, which includes an introduction to motives of quadrics by A. Vishik, with various applications, notably to the splitting patterns of quadratic forms, papers by O. Izhboldin and N. Karpenko on Chow groups of quadrics and their stable birational equivalence, with application to the construction of fields with u-invariant 9, and a contribution in French by B. Kahn which lays out a general framework for the computation of the unramified cohomology groups of quadrics and other cellular varieties.
| Author | : Edwin Weiss |
| Publisher | : Courier Corporation |
| Total Pages | : 308 |
| Release | : 2012-01-27 |
| Genre | : Mathematics |
| ISBN | : 048615436X |
Download Algebraic Number Theory Book in PDF, ePub and Kindle
Ideal either for classroom use or as exercises for mathematically minded individuals, this text introduces elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.
| Author | : Helmut Koch |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 390 |
| Release | : 2000 |
| Genre | : Mathematics |
| ISBN | : 9780821820544 |
Download Number Theory Book in PDF, ePub and Kindle
Algebraic number theory is one of the most refined creations in mathematics. It has been developed by some of the leading mathematicians of this and previous centuries. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory. Following the example set for us by Kronecker, Weber, Hilbert and Artin, algebraic functions are handled here on an equal footing with algebraic numbers. This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially clear in the case where the ground field is a finite field. On the other hand, in this way one obtains an introduction to the theory of 'higher congruences' as an important element of 'arithmetic geometry'. Early chapters discuss topics in elementary number theory, such as Minkowski's geometry of numbers, public-key cryptography and a short proof of the Prime Number Theorem, following Newman and Zagier. Next, some of the tools of algebraic number theory are introduced, such as ideals, discriminants and valuations. These results are then applied to obtain results about function fields, including a proof of the Riemann-Roch Theorem and, as an application of cyclotomic fields, a proof of the first case of Fermat's Last Theorem. There are a detailed exposition of the theory of Hecke $L$-series, following Tate, and explicit applications to number theory, such as the Generalized Riemann Hypothesis. Chapter 9 brings together the earlier material through the study of quadratic number fields. Finally, Chapter 10 gives an introduction to class field theory. The book attempts as much as possible to give simple proofs. It can be used by a beginner in algebraic number theory who wishes to see some of the true power and depth of the subject. The book is suitable for two one-semester courses, with the first four chapters serving to develop the basic material. Chapters 6 through 9 could be used on their own as a second semester course.
