Algebraic Theory Of Quadratic Numbers

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Algebraic Theory of Quadratic Numbers

Author : Mak Trifković
Publisher : Springer Science & Business Media
ISBN 13 : 1461477174
Total Pages : 206 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Algebraic Theory of Quadratic Numbers by : Mak Trifković

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.

Bilinear Algebra

Author : Kazimierz Szymiczek
Publisher : Routledge
ISBN 13 : 1351464205
Total Pages : 508 pages
Book Rating : 4.3/5 (514 download)

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Book Synopsis Bilinear Algebra by : Kazimierz Szymiczek

Download or read book Bilinear Algebra written by Kazimierz Szymiczek and published by Routledge. This book was released on 2017-11-22 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic

Author : J. L. Lehman
Publisher : American Mathematical Soc.
ISBN 13 : 1470447371
Total Pages : 394 pages
Book Rating : 4.4/5 (74 download)

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

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.

Algebraic Theory of Quadratic Numbers

Author : Mak Trifkovi
Publisher :
ISBN 13 : 9781461477181
Total Pages : 212 pages
Book Rating : 4.4/5 (771 download)

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Book Synopsis Algebraic Theory of Quadratic Numbers by : Mak Trifkovi

Download or read book Algebraic Theory of Quadratic Numbers written by Mak Trifkovi and published by . This book was released on 2013-08-31 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Algebraic and Geometric Theory of Quadratic Forms

Author : Richard S. Elman
Publisher : American Mathematical Soc.
ISBN 13 : 9780821873229
Total Pages : 456 pages
Book Rating : 4.8/5 (732 download)

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Book Synopsis The Algebraic and Geometric Theory of Quadratic Forms by : Richard S. Elman

Download or read book The Algebraic and Geometric Theory of Quadratic Forms written by Richard S. Elman and published by American Mathematical Soc.. This book was released on 2008-07-15 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Quadratic Number Fields

Author : Franz Lemmermeyer
Publisher : Springer Nature
ISBN 13 : 3030786528
Total Pages : 348 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Quadratic Number Fields by : Franz Lemmermeyer

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.

Classical Theory of Algebraic Numbers

Author : Paulo Ribenboim
Publisher : Springer Science & Business Media
ISBN 13 : 0387216901
Total Pages : 676 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Classical Theory of Algebraic Numbers by : Paulo Ribenboim

Download or read book Classical Theory of Algebraic Numbers written by Paulo Ribenboim and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 676 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Geometric Methods in the Algebraic Theory of Quadratic Forms

Author : Oleg T. Izhboldin
Publisher : Springer
ISBN 13 : 3540409904
Total Pages : 198 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Geometric Methods in the Algebraic Theory of Quadratic Forms by : Oleg T. Izhboldin

Download or read book Geometric Methods in the Algebraic Theory of Quadratic Forms written by Oleg T. Izhboldin and published by Springer. This book was released on 2004-02-07 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Algebraic Number Theory

Author : Edwin Weiss
Publisher : Courier Corporation
ISBN 13 : 048615436X
Total Pages : 308 pages
Book Rating : 4.4/5 (861 download)

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Book Synopsis Algebraic Number Theory by : Edwin Weiss

Download or read book Algebraic Number Theory written by Edwin Weiss and published by Courier Corporation. This book was released on 2012-01-27 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Lectures on the Theory of Algebraic Numbers

Author : E. T. Hecke
Publisher : Springer Science & Business Media
ISBN 13 : 1475740921
Total Pages : 251 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Lectures on the Theory of Algebraic Numbers by : E. T. Hecke

Download or read book Lectures on the Theory of Algebraic Numbers written by E. T. Hecke and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: . . . if one wants to make progress in mathematics one should study the masters not the pupils. N. H. Abel Heeke was certainly one of the masters, and in fact, the study of Heeke L series and Heeke operators has permanently embedded his name in the fabric of number theory. It is a rare occurrence when a master writes a basic book, and Heeke's Lectures on the Theory of Algebraic Numbers has become a classic. To quote another master, Andre Weil: "To improve upon Heeke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task. " We have tried to remain as close as possible to the original text in pre serving Heeke's rich, informal style of exposition. In a very few instances we have substituted modern terminology for Heeke's, e. g. , "torsion free group" for "pure group. " One problem for a student is the lack of exercises in the book. However, given the large number of texts available in algebraic number theory, this is not a serious drawback. In particular we recommend Number Fields by D. A. Marcus (Springer-Verlag) as a particularly rich source. We would like to thank James M. Vaughn Jr. and the Vaughn Foundation Fund for their encouragement and generous support of Jay R. Goldman without which this translation would never have appeared. Minneapolis George U. Brauer July 1981 Jay R.

The Algebraic Theory of Quadratic Forms

Author : Tsit-Yuen Lam
Publisher : Addison-Wesley
ISBN 13 : 9780805356663
Total Pages : 344 pages
Book Rating : 4.3/5 (566 download)

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Book Synopsis The Algebraic Theory of Quadratic Forms by : Tsit-Yuen Lam

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:

Number Theory

Author : Helmut Koch
Publisher : American Mathematical Soc.
ISBN 13 : 9780821820544
Total Pages : 390 pages
Book Rating : 4.8/5 (25 download)

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Book Synopsis Number Theory by : Helmut Koch

Download or read book Number Theory written by Helmut Koch and published by American Mathematical Soc.. This book was released on 2000 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Algebraic Theory of Numbers

Author : Pierre Samuel
Publisher : Dover Books on Mathematics
ISBN 13 : 9780486466668
Total Pages : 0 pages
Book Rating : 4.4/5 (666 download)

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Book Synopsis Algebraic Theory of Numbers by : Pierre Samuel

Download or read book Algebraic Theory of Numbers written by Pierre Samuel and published by Dover Books on Mathematics. This book was released on 2008 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic number theory introduces students to new algebraic notions as well as related concepts: groups, rings, fields, ideals, quotient rings, and quotient fields. This text covers the basics, from divisibility theory in principal ideal domains to the unit theorem, finiteness of the class number, and Hilbert ramification theory. 1970 edition.

Introduction to Quadratic Forms over Fields

Author : Tsit-Yuen Lam
Publisher : American Mathematical Soc.
ISBN 13 : 0821810952
Total Pages : 577 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Introduction to Quadratic Forms over Fields by : Tsit-Yuen Lam

Download or read book Introduction to Quadratic Forms over Fields written by Tsit-Yuen Lam and published by American Mathematical Soc.. This book was released on 2005 with total page 577 pages. Available in PDF, EPUB and Kindle. Book excerpt: This new version of the author's prizewinning book, Algebraic Theory of Quadratic Forms (W. A. Benjamin, Inc., 1973), gives a modern and self-contained introduction to the theory of quadratic forms over fields of characteristic different from two. Starting with few prerequisites beyond linear algebra, the author charts an expert course from Witt's classical theory of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of formally real fields, and structural theorems on Witt rings, to the theory of Pfister forms, function fields, and field invariants. These main developments are seamlessly interwoven with excursions into Brauer-Wall groups, local and global fields, trace forms, Galois theory, and elementary algebraic K-theory, to create a uniquely original treatment of quadratic form theory over fields. Two new chapters totaling more than 100 pages have been added to the earlier incarnation of this book to take into account some of the newer results and more recent viewpoints in the area. As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.

Quadratic Irrationals

Author : Franz Halter-Koch
Publisher : CRC Press
ISBN 13 : 1466591846
Total Pages : 431 pages
Book Rating : 4.4/5 (665 download)

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Book Synopsis Quadratic Irrationals by : Franz Halter-Koch

Download or read book Quadratic Irrationals written by Franz Halter-Koch and published by CRC Press. This book was released on 2013-06-17 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quadratic Irrationals: An Introduction to Classical Number Theory gives a unified treatment of the classical theory of quadratic irrationals. Presenting the material in a modern and elementary algebraic setting, the author focuses on equivalence, continued fractions, quadratic characters, quadratic orders, binary quadratic forms, and class groups.T

Problems in Algebraic Number Theory

Author : M. Ram Murty
Publisher : Springer Science & Business Media
ISBN 13 : 0387269983
Total Pages : 354 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Problems in Algebraic Number Theory by : M. Ram Murty

Download or read book Problems in Algebraic Number Theory written by M. Ram Murty and published by Springer Science & Business Media. This book was released on 2005-09-28 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problems are systematically arranged to reveal the evolution of concepts and ideas of the subject Includes various levels of problems - some are easy and straightforward, while others are more challenging All problems are elegantly solved

A Conversational Introduction to Algebraic Number Theory

Author : Paul Pollack
Publisher : American Mathematical Soc.
ISBN 13 : 1470436531
Total Pages : 329 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis A Conversational Introduction to Algebraic Number Theory by : Paul Pollack

Download or read book A Conversational Introduction to Algebraic Number Theory written by Paul Pollack and published by American Mathematical Soc.. This book was released on 2017-08-01 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gauss famously referred to mathematics as the “queen of the sciences” and to number theory as the “queen of mathematics”. This book is an introduction to algebraic number theory, meaning the study of arithmetic in finite extensions of the rational number field Q . Originating in the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This book lays out basic results, including the three “fundamental theorems”: unique factorization of ideals, finiteness of the class number, and Dirichlet's unit theorem. While these theorems are by now quite classical, both the text and the exercises allude frequently to more recent developments. In addition to traversing the main highways, the book reveals some remarkable vistas by exploring scenic side roads. Several topics appear that are not present in the usual introductory texts. One example is the inclusion of an extensive discussion of the theory of elasticity, which provides a precise way of measuring the failure of unique factorization. The book is based on the author's notes from a course delivered at the University of Georgia; pains have been taken to preserve the conversational style of the original lectures.