Quadratic Number Theory An Invitation To Algebraic Methods In The Higher Arithmetic PDF Download
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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 | : 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 | : Harold M. Edwards |
Publisher | : American Mathematical Soc. |
Total Pages | : 228 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 9780821844397 |
Download Higher Arithmetic Book in PDF, ePub and Kindle
Among the topics featured in this textbook are: congruences; the fundamental theorem of arithmetic; exponentiation and orders; primality testing; the RSA cipher system; polynomials; modules of hypernumbers; signatures of equivalence classes; and the theory of binary quadratic forms. The book contains exercises with answers.
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 | : Harvey Cohn |
Publisher | : Springer Science & Business Media |
Total Pages | : 344 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461299500 |
Download A Classical Invitation to Algebraic Numbers and Class Fields Book in PDF, ePub and Kindle
"Artin's 1932 Göttingen Lectures on Class Field Theory" and "Connections between Algebrac Number Theory and Integral Matrices"
Author | : Franz Halter-Koch |
Publisher | : CRC Press |
Total Pages | : 595 |
Release | : 2020-05-04 |
Genre | : Mathematics |
ISBN | : 0429014678 |
Download An Invitation To Algebraic Numbers And Algebraic Functions Book in PDF, ePub and Kindle
The author offers a thorough presentation of the classical theory of algebraic numbers and algebraic functions which both in its conception and in many details differs from the current literature on the subject. The basic features are: Field-theoretic preliminaries and a detailed presentation of Dedekind’s ideal theory including non-principal orders and various types of class groups; the classical theory of algebraic number fields with a focus on quadratic, cubic and cyclotomic fields; basics of the analytic theory including the prime ideal theorem, density results and the determination of the arithmetic by the class group; a thorough presentation of valuation theory including the theory of difference, discriminants, and higher ramification. The theory of function fields is based on the ideal and valuation theory developed before; it presents the Riemann-Roch theorem on the basis of Weil differentials and highlights in detail the connection with classical differentials. The theory of congruence zeta functions and a proof of the Hasse-Weil theorem represent the culminating point of the volume. The volume is accessible with a basic knowledge in algebra and elementary number theory. It empowers the reader to follow the advanced number-theoretic literature, and is a solid basis for the study of the forthcoming volume on the foundations and main results of class field theory. Key features: • A thorough presentation of the theory of Algebraic Numbers and Algebraic Functions on an ideal and valuation-theoretic basis. • Several of the topics both in the number field and in the function field case were not presented before in this context. • Despite presenting many advanced topics, the text is easily readable. Franz Halter-Koch is professor emeritus at the university of Graz. He is the author of “Ideal Systems” (Marcel Dekker,1998), “Quadratic Irrationals” (CRC, 2013), and a co-author of “Non-Unique Factorizations” (CRC 2006).
Author | : H. Davenport |
Publisher | : Cambridge University Press |
Total Pages | : 248 |
Release | : 1999-12-09 |
Genre | : Mathematics |
ISBN | : 9780521634465 |
Download The Higher Arithmetic Book in PDF, ePub and Kindle
Seventh edition of a classic elementary number theory book.
Author | : Burton W Jones |
Publisher | : American Mathematical Soc. |
Total Pages | : 212 |
Release | : 1950-12-31 |
Genre | : Forms, Binary |
ISBN | : 1614440107 |
Download The Arithmetic Theory of Quadratic Forms Book in PDF, ePub and Kindle
This monograph presents the central ideas of the arithmetic theory of quadratic forms in self-contained form, assuming only knowledge of the fundamentals of matric theory and the theory of numbers. Pertinent concepts of p -adic numbers and quadratic ideals are introduced. It would have been possible to avoid these concepts, but the theory gains elegance as well as breadth by the introduction of such relationships. Some results, and many of the methods, are here presented for the first time. The development begins with the classical theory in the field of reals from the point of view of representation theory; for in these terms, many of the later objectives and methods may be revealed. The successive chapters gradually narrow the fields and rings until one has the tools at hand to deal with the classical problems in the ring of rational integers. The analytic theory of quadratic forms is not dealt with because of the delicate analysis involved. However, some of the more important results are stated and references are given.
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 | : Andrew Granville |
Publisher | : American Mathematical Society |
Total Pages | : 587 |
Release | : 2020-09-23 |
Genre | : Mathematics |
ISBN | : 1470463709 |
Download Number Theory Revealed: A Masterclass Book in PDF, ePub and Kindle
Number Theory Revealed: A Masterclass acquaints enthusiastic students with the “Queen of Mathematics”. The text offers a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod $p$ and modern twists on traditional questions like the values represented by binary quadratic forms, the anatomy of integers, and elliptic curves. This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction, highlighting beautiful developments and inspiring other subjects in mathematics (like algebra). This allows instructors to tailor a course suited to their own (and their students') interests. There are new yet accessible topics like the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, a new proof of Mordell's Theorem for congruent elliptic curves, and a discussion of the $abc$-conjecture including its proof for polynomials. About the Author: Andrew Granville is the Canada Research Chair in Number Theory at the University of Montreal and professor of mathematics at University College London. He has won several international writing prizes for exposition in mathematics, including the 2008 Chauvenet Prize and the 2019 Halmos-Ford Prize, and is the author of Prime Suspects (Princeton University Press, 2019), a beautifully illustrated graphic novel murder mystery that explores surprising connections between the anatomies of integers and of permutations.