The present book was conceived as an introduction for the user of universal algebra, rather than a handbook for the specialist, but when the first edition appeared in 1965, there were practically no other books entir ly devoted to the subject, whether introductory or specialized. Today the specialist in the field is well provided for, but there is still a demand for an introduction to the subject to suit the user, and this seemed to justify a reissue of the book. Naturally some changes have had to be made; in particular, I have corrected all errors that have been brought to my notice. Besides errors, some obscurities in the text have been removed and the references brought up to date. I should like to express my thanks to a number of correspondents for their help, in particular C. G. d'Ambly, W. Felscher, P. Goralcik, P. J. Higgins, H.-J. Hoehnke, J. R. Isbell, A. H. Kruse, E. J. Peake, D. Suter, J. S. Wilson. But lowe a special debt to G. M. Bergman, who has provided me with extensive comments. particularly on Chapter VII and the supplementary chapters. I have also con sulted reviews of the first edition, as well as the Italian and Russian translations."

When I began to write this book, I originally had in mind the needs of university students in their first year. May aim was to keep the mathematics simple. No advanced techniques are used and there are no complicated applications. The emphasis is on an understanding of the basic ideas and problems which require expertise but do not contribute to this understanding are not discussed. How ever, the presentation is more sophisticated than might be considered appropri ate for someone with no previous knowledge of the subject so that, although it is developed from the beginning, some previous acquaintance with the elements of the subject would be an advantage. In addition, some familiarity with element ary calculus is assumed but not with the elementary theory of differential equations, although knowledge of the latter would again be an advantage. It is my opinion that mechanics is best introduced through the motion of a particle, with rigid body problems left until the subject is more fully developed. However, some experienced mathematicians consider that no introduction is complete without a discussion of rigid body mechanics. Conventional accounts of the subject invariably include such a discussion, but with the problems restricted to two-dimensional ones in the books which claim to be elementary. The mechanics of rigid bodies is therefore included but there is no separate discussion of the theory in two dimensions."

This is a collection of research-oriented monographs, reports, and notes arising from lectures and seminars on the Weil representation, the Maslov index, and the Theta series. It is good contribution to the international scientific community, particularly for researchers and graduate students in the field.

Berkeley Lectures on p-adic Geometry presents an important breakthrough in arithmetic geometry. In 2014, leading mathematician Peter Scholze delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, Scholze introduced the concept of "diamonds," which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. In this book, Peter Scholze and Jared Weinstein show that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. This book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained. Berkeley Lectures on p-adic Geometry will be a useful resource for students and scholars working in arithmetic geometry and number theory.

In the summer quarter of 1949, I taught a ten-weeks introductory course on number theory at the University of Chicago; it was announced in the catalogue as "Alge bra 251." What made it possible, in the form which I had planned for it, was the fact that Max Rosenlicht, now of the University of California at Berkeley, was then my assistant. According to his recollection, "this was the first and last time, in the his tory of the Chicago department of mathematics, that an assistant worked for his salary." The course consisted of two lectures a week, supplemented by a weekly "laboratory period" where students were given exercises which they were. asked to solve under Max's supervision and (when necessary) with his help. This idea was borrowed from the "Praktikum" of German universi ties. Being alien to the local tradition, it did not work out as well as I had hoped, and student attendance at the problem sessions so on became desultory. v vi Weekly notes were written up by Max Rosenlicht and issued week by week to the students. Rather than a literal reproduction of the course, they should be regarded as its skeleton; they were supplemented by references to stan dard text-books on algebra. Max also contributed by far the larger part of the exercises. None of, this was meant for publication."

This book presents a self-contained introduction to H.M. Stark 's remarkable conjectures about the leading term of the Taylor expansion of Artin 's L-functions at s=0. These conjectures can be viewed as a vast generalization of Dirichlet 's class number formula and Kronecker 's limit formula. They provide an unexpected contribution to Hilbert 's 12th problem on the generalization of class fields by the values of transcendental functions. This volume belongs on the shelf of every mathematics library.

Building on the success of the first edition, An Introduction to Number Theory with Cryptography, Second Edition, increases coverage of the popular and important topic of cryptography, integrating it with traditional topics in number theory. The authors have written the text in an engaging style to reflect number theory's increasing popularity. The book is designed to be used by sophomore, junior, and senior undergraduates, but it is also accessible to advanced high school students and is appropriate for independent study. It includes a few more advanced topics for students who wish to explore beyond the traditional curriculum. Features of the second edition include Over 800 exercises, projects, and computer explorations Increased coverage of cryptography, including Vigenere, Stream, Transposition,and Block ciphers, along with RSA and discrete log-based systems "Check Your Understanding" questions for instant feedback to students New Appendices on "What is a proof?" and on Matrices Select basic (pre-RSA) cryptography now placed in an earlier chapter so that the topic can be covered right after the basic material on congruences Answers and hints for odd-numbered problems About the Authors: Jim Kraft received his Ph.D. from the University of Maryland in 1987 and has published several research papers in algebraic number theory. His previous teaching positions include the University of Rochester, St. Mary's College of California, and Ithaca College, and he has also worked in communications security. Dr. Kraft currently teaches mathematics at the Gilman School. Larry Washington received his Ph.D. from Princeton University in 1974 and has published extensively in number theory, including books on cryptography (with Wade Trappe), cyclotomic fields, and elliptic curves. Dr. Washington is currently Professor of Mathematics and Distinguished Scholar-Teacher at the University of Maryland.

This book is a comprehensive treatise on the partial toroidal and minimal compactifications of the ordinary loci of PEL-type Shimura varieties and Kuga families, and on the canonical and subcanonical extensions of automorphic bundles. The results in this book serve as the logical foundation of several recent developments in the theory of p-adic automorphic forms; and of the author's work with Harris, Taylor, and Thorne on the construction of Galois representations without any polarizability conditions, which is a major breakthrough in the Langlands program.This book is important for active researchers and graduate students who need to understand the above-mentioned recent works, and is written with such users of the theory in mind, providing plenty of explanations and background materials, which should be helpful for people working in similar areas. It also contains precise internal and external references, and an index of notation and terminologies. These are useful for readers to quickly locate materials they need.

Auf breiter fachlicher Ebene werden in dem Lehrbuch einfache elementare zahlentheoretische Inhalte besprochen, aber auch Stoffkomplexe aus der analytischen und algebraischen Zahlentheorie dargestellt. Das Buch bietet so auf uberschaubaren mathematischen Niveau einen Einstieg in ausgewahlte Themen der Zahlentheorie. Samtliche Kapitel enthalten umfassend Beispiele, UEbungsaufgaben mit Loesungen, Abbildungen und ausfuhrlich durchgerechnete Beweise, so dass es sich sehr gut zur Prufungsvorbereitung eignet.

Designed as a self-contained account of a number of key algorithmic problems and their solutions for linear algebraic groups, this book combines in one single text both an introduction to the basic theory of linear algebraic groups and a substantial collection of useful algorithms. Computation with Linear Algebraic Groups offers an invaluable guide to graduate students and researchers working in algebraic groups, computational algebraic geometry, and computational group theory, as well as those looking for a concise introduction to the theory of linear algebraic groups.

This textbook offers an introduction to the theory of Drinfeld modules, mathematical objects that are fundamental to modern number theory. After the first two chapters conveniently recalling prerequisites from abstract algebra and non-Archimedean analysis, Chapter 3 introduces Drinfeld modules and the key notions of isogenies and torsion points. Over the next four chapters, Drinfeld modules are studied in settings of various fields of arithmetic importance, culminating in the case of global fields. Throughout, numerous number-theoretic applications are discussed, and the analogies between classical and function field arithmetic are emphasized. Drinfeld Modules guides readers from the basics to research topics in function field arithmetic, assuming only familiarity with graduate-level abstract algebra as prerequisite. With exercises of varying difficulty included in each section, the book is designed to be used as the primary textbook for a graduate course on the topic, and may also provide a supplementary reference for courses in algebraic number theory, elliptic curves, and related fields. Furthermore, researchers in algebra and number theory will appreciate it as a self-contained reference on the topic.

This is the sixth annual volume of papers based on the outstanding lectures given at the Seminaire de Theorie des Nombres de Paris. The results presented in 1985-86 by an international group of mathematicians reflect the most recent work in many areas of number theory.

L-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy to read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2, R) and GL(3, R), and then for the general case of GL(n, R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.

"Number Theory: A Lively Introduction with Proofs, Applications, and Stories," is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications.

Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. Many theorems are preceded by "Numerical Proof Previews," which are numerical examples that will help give students a concrete understanding of both the statements of the theorems and the ideas behind their proofs, before the statement and proof are formalized in more abstract terms. In addition, many applications of number theory are explained in detail throughout the text, including some that have rarely (if ever) appeared in textbooks.

A unique feature of the book is that every chapter includes a "math myth," a fictional story that introduces an important number theory topic in a friendly, inviting manner. Many of the exercise sets include in-depth "Explorations," in which a series of exercises develop a topic that is related to the material in the section.

Introduction to Number Theory is a classroom-tested, student-friendly text that covers a diverse array of number theory topics, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments such as cryptography, the theory of elliptic curves, and the negative solution of Hilbert's tenth problem. The authors illustrate the connections between number theory and other areas of mathematics, including algebra, analysis, and combinatorics. They also describe applications of number theory to real-world problems, such as congruences in the ISBN system, modular arithmetic and Euler's theorem in RSA encryption, and quadratic residues in the construction of tournaments. Ideal for a one- or two-semester undergraduate-level course, this Second Edition: Features a more flexible structure that offers a greater range of options for course design Adds new sections on the representations of integers and the Chinese remainder theorem Expands exercise sets to encompass a wider variety of problems, many of which relate number theory to fields outside of mathematics (e.g., music) Provides calculations for computational experimentation using SageMath, a free open-source mathematics software system, as well as Mathematica (R) and Maple (TM), online via a robust, author-maintained website Includes a solutions manual with qualifying course adoption By tackling both fundamental and advanced subjects-and using worked examples, numerous exercises, and popular software packages to ensure a practical understanding-Introduction to Number Theory, Second Edition instills a solid foundation of number theory knowledge.

Understanding maths has never been easier. Combining bold, elegant graphics with easy-to-understand text, Simply Maths is the perfect introduction to the subject for those who are short of time but hungry for knowledge. Covering more than 90 key mathematical concepts from prime numbers and fractions to quadratic equations and probability experiments, each pared-back, single-page entry explains the concept more clearly than ever before. Organized by major themes - number theory and systems; calculations; geometry; algebra; graphs; ratio and proportion; measurement; probability and statistics; and calculus - entries explain the essentials of each key mathematical theory with simple clarity and for ease of understanding. Whether you are studying maths at school or college, or simply want a jargon-free overview of the subject, this indispensable guide is packed with everything you need to understand the basics quickly and easily.