Quadratic Number Theory
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Quadratic Number Theory
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Author : J. L. Lehman
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-02-13
Quadratic Number Theory written by J. L. Lehman and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-02-13 with Mathematics categories.
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
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Author : Mak Trifković
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-09-14
Algebraic Theory Of Quadratic Numbers written by Mak Trifković and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-09-14 with Mathematics categories.
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.
Quadratic Number Fields
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Author : Franz Lemmermeyer
language : en
Publisher: Springer Nature
Release Date : 2021-09-18
Quadratic Number Fields written by Franz Lemmermeyer and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-18 with Mathematics categories.
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.
Quadratic Irrationals
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Author : Franz Halter-Koch
language : en
Publisher: CRC Press
Release Date : 2013-06-17
Quadratic Irrationals written by Franz Halter-Koch and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-17 with Mathematics categories.
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
Algebraic Number Theory
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Author : Frazer Jarvis
language : en
Publisher: Springer
Release Date : 2014-06-23
Algebraic Number Theory written by Frazer Jarvis and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-06-23 with Mathematics categories.
This undergraduate textbook provides an approachable and thorough introduction to the topic of algebraic number theory, taking the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.
Binary Quadratic Forms
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Author : Duncan A. Buell
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Binary Quadratic Forms written by Duncan A. Buell and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
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 quadraticforms 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.
Fundamentals Of Number Theory
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Author : William J. LeVeque
language : en
Publisher: Courier Corporation
Release Date : 2014-01-05
Fundamentals Of Number Theory written by William J. LeVeque and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-05 with Mathematics categories.
This excellent textbook introduces the basics of number theory, incorporating the language of abstract algebra. A knowledge of such algebraic concepts as group, ring, field, and domain is not assumed, however; all terms are defined and examples are given — making the book self-contained in this respect. The author begins with an introductory chapter on number theory and its early history. Subsequent chapters deal with unique factorization and the GCD, quadratic residues, number-theoretic functions and the distribution of primes, sums of squares, quadratic equations and quadratic fields, diophantine approximation, and more. Included are discussions of topics not always found in introductory texts: factorization and primality of large integers, p-adic numbers, algebraic number fields, Brun's theorem on twin primes, and the transcendence of e, to mention a few. Readers will find a substantial number of well-chosen problems, along with many notes and bibliographical references selected for readability and relevance. Five helpful appendixes — containing such study aids as a factor table, computer-plotted graphs, a table of indices, the Greek alphabet, and a list of symbols — and a bibliography round out this well-written text, which is directed toward undergraduate majors and beginning graduate students in mathematics. No post-calculus prerequisite is assumed. 1977 edition.
Algebraic Number Theory And Fermat S Last Theorem
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Author : Ian Stewart
language : en
Publisher: CRC Press
Release Date : 2001-12-12
Algebraic Number Theory And Fermat S Last Theorem written by Ian Stewart and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-12-12 with Mathematics categories.
First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat's Last Theorem. Intended as a upper level textbook, it
Quadratic Number Theory
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Author : James Larry Lehman
language : en
Publisher: MAA Press
Release Date : 2019
Quadratic Number Theory written by James Larry Lehman and has been published by MAA Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with MATHEMATICS categories.
"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."--Site web de l'éditeur.