Advertisement
Algebra and Number Theory: The Unsung Heroes of Modern Industry
By Dr. Evelyn Reed, PhD
Dr. Evelyn Reed holds a PhD in Applied Mathematics from MIT and has over 15 years of experience in cryptography and data security, working with leading tech companies like Google and Amazon.
Published by: TechReview Publications – A leading publisher of peer-reviewed articles and industry reports, renowned for its rigorous editorial process and commitment to accuracy in the technology sector.
Edited by: Dr. Michael Chen, A seasoned editor with over 20 years of experience in technical publishing, specializing in mathematics and computer science.
Keywords: algebra and number theory, cryptography, data security, error correction codes, optimization, machine learning, industry applications, number theory applications, algebraic structures.
Abstract: This article explores the surprising and pervasive impact of algebra and number theory on various industries. While often perceived as abstract mathematical fields, their fundamental principles underpin crucial technologies in areas ranging from cybersecurity to machine learning and optimization. We will delve into specific examples, highlighting the practical applications and future potential of algebra and number theory.
1. The Foundation: Understanding Algebra and Number Theory
Algebra and number theory, seemingly disparate fields, are intrinsically linked. Number theory focuses on the properties of integers, while algebra provides the tools – such as groups, rings, and fields – to study them systematically. This powerful combination allows for the development of sophisticated algorithms and mathematical frameworks with far-reaching consequences. The elegance and rigor of algebra and number theory provide solutions to complex problems that other approaches struggle with.
2. Cryptography: Securing Our Digital World
One of the most impactful applications of algebra and number theory is in cryptography. The security of online transactions, sensitive data, and communication networks relies heavily on these mathematical principles. Public-key cryptography, for example, utilizes the difficulty of factoring large numbers (a problem in number theory) and the properties of modular arithmetic (an algebraic concept) to ensure secure communication. Algorithms like RSA and ECC (Elliptic Curve Cryptography) are prime examples of this application. The strength of these cryptographic systems directly stems from the computational hardness of problems within algebra and number theory.
3. Error Correction Codes: Ensuring Data Integrity
In data transmission and storage, errors can occur. Error correction codes, vital for reliable communication in various contexts from satellite transmissions to hard drive storage, rely heavily on algebraic structures like finite fields. These codes use sophisticated mathematical techniques to detect and correct errors, ensuring data integrity. This is crucial in scenarios where data loss is unacceptable, such as in medical imaging, financial transactions, and space exploration. The efficient implementation of these codes heavily depends on the theoretical insights provided by algebra and number theory.
4. Optimization and Machine Learning: Improving Efficiency and Accuracy
Beyond cryptography and error correction, algebra and number theory play a significant role in optimization problems. Linear algebra, a cornerstone of algebra, is fundamental to many machine learning algorithms. Techniques like linear regression and support vector machines rely heavily on matrix operations and vector spaces. Furthermore, advancements in optimization algorithms often leverage concepts from number theory to find efficient solutions to complex problems arising in areas like logistics, supply chain management, and resource allocation.
5. The Future of Algebra and Number Theory in Industry
The applications of algebra and number theory are constantly expanding. Research in areas like post-quantum cryptography, which aims to develop cryptographic systems resistant to attacks from quantum computers, relies heavily on advanced algebraic structures and number-theoretic problems. The development of more efficient machine learning algorithms and the exploration of new optimization techniques also benefit significantly from advancements in these fundamental mathematical fields.
Conclusion
Algebra and number theory, despite their abstract nature, are not confined to the realm of pure mathematics. They serve as the bedrock for numerous critical technologies across various industries. Their impact on cybersecurity, data integrity, and optimization demonstrates the profound and often overlooked contribution of these mathematical fields to our modern world. Continued research and innovation in algebra and number theory will undoubtedly lead to further breakthroughs with far-reaching implications for industry and society as a whole.
Frequently Asked Questions (FAQs)
1. What is the difference between algebra and number theory? Algebra deals with abstract structures and operations, while number theory focuses specifically on the properties of integers. They often intersect and complement each other.
2. How is algebra used in cryptography? Algebra provides the framework for designing cryptographic systems, particularly public-key cryptography, through concepts like modular arithmetic and groups.
3. What are some real-world examples of error correction codes? DVDs, Blu-ray discs, and satellite communications all utilize error correction codes to ensure data integrity.
4. How does number theory contribute to machine learning? While not as directly apparent, number theory can impact the efficiency of algorithms used in machine learning, particularly in optimization problems.
5. What is post-quantum cryptography? It's the development of cryptographic systems that are secure against attacks from quantum computers, relying heavily on advanced algebraic structures.
6. Is there a shortage of mathematicians with expertise in algebra and number theory? Yes, there is a significant demand for skilled mathematicians with expertise in these areas, especially in the tech industry.
7. How can I learn more about algebra and number theory? Start with introductory textbooks and online courses, and then explore more specialized literature based on your interests.
8. What are the ethical implications of using algebra and number theory in technology? The ethical considerations center around the responsible use of encryption and data security technologies to prevent misuse and protect privacy.
9. Are there any open problems in algebra and number theory with significant industry implications? Yes, many open problems exist, especially in areas like factoring large numbers and developing more efficient algorithms for cryptographic applications.
Related Articles:
1. "Elliptic Curve Cryptography: A Primer": Explains the fundamentals of ECC and its applications in cryptography.
2. "The RSA Algorithm: A Deep Dive": A detailed exploration of the RSA algorithm and its mathematical underpinnings.
3. "Finite Fields and their Applications in Coding Theory": Covers the use of finite fields in the construction of error correction codes.
4. "Linear Algebra for Machine Learning": Introduces the essential concepts of linear algebra relevant to machine learning algorithms.
5. "Introduction to Number Theory and Cryptography": A beginner-friendly overview of the connection between number theory and cryptography.
6. "Optimization Algorithms and their Applications": Discusses various optimization techniques and their applications in different fields.
7. "Post-Quantum Cryptography: Challenges and Opportunities": Explores the current research and challenges in post-quantum cryptography.
8. "The Mathematics of Data Security": Explores the mathematical foundations of various data security techniques.
9. "Algebraic Structures and their Applications in Computer Science": Examines the application of various algebraic structures in computer science problems.
| algebra and number theory: Algebra and Number Theory Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, 2011-07-15 Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines—linear algebra, abstract algebra, and number theory—into one comprehensive and fluid presentation, facilitating a deeper understanding of the topic and improving readers' retention of the main concepts. The book begins with an introduction to the elements of set theory. Next, the authors discuss matrices, determinants, and elements of field theory, including preliminary information related to integers and complex numbers. Subsequent chapters explore key ideas relating to linear algebra such as vector spaces, linear mapping, and bilinear forms. The book explores the development of the main ideas of algebraic structures and concludes with applications of algebraic ideas to number theory. Interesting applications are provided throughout to demonstrate the relevance of the discussed concepts. In addition, chapter exercises allow readers to test their comprehension of the presented material. Algebra and Number Theory is an excellent book for courses on linear algebra, abstract algebra, and number theory at the upper-undergraduate level. It is also a valuable reference for researchers working in different fields of mathematics, computer science, and engineering as well as for individuals preparing for a career in mathematics education. |
| algebra and number theory: Algebraic Number Theory Edwin Weiss, 2012-01-27 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. |
| algebra and number theory: An Introduction to Commutative Algebra and Number Theory Sukumar Das Adhikari, 2001-11 This is an elementary introduction to algebra and number theory. The text begins by a review of groups, rings, and fields. The algebra portion addresses polynomial rings, UFD, PID, and Euclidean domains, field extensions, modules, and Dedckind domains. The number theory portion reviews elementary congruence, quadratic reciprocity, algebraic number fields, and a glimpse into the various aspects of that subject. This book could be used as a one semester course in graduate mathematics. |
| algebra and number theory: Algebraic Number Theory Edwin Weiss, 1998-01-01 Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis). Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract techniques constitute the primary focus. Topics include introductory materials on elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields. Subjects correspond to those usually covered in a one-semester, graduate level course in algebraic number theory, making this book ideal either for classroom use or as a stimulating series of exercises for mathematically minded individuals. |
| algebra and number theory: Problems in Algebraic Number Theory M. Ram Murty, Jody (Indigo) Esmonde, 2005-09-28 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 |
| algebra and number theory: Algebraic Number Theory and Fermat's Last Theorem Ian Stewart, David Tall, 2001-12-12 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 |
| algebra and number theory: Algebra and Number Theory Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, 2011-07-15 Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines—linear algebra, abstract algebra, and number theory—into one comprehensive and fluid presentation, facilitating a deeper understanding of the topic and improving readers' retention of the main concepts. The book begins with an introduction to the elements of set theory. Next, the authors discuss matrices, determinants, and elements of field theory, including preliminary information related to integers and complex numbers. Subsequent chapters explore key ideas relating to linear algebra such as vector spaces, linear mapping, and bilinear forms. The book explores the development of the main ideas of algebraic structures and concludes with applications of algebraic ideas to number theory. Interesting applications are provided throughout to demonstrate the relevance of the discussed concepts. In addition, chapter exercises allow readers to test their comprehension of the presented material. Algebra and Number Theory is an excellent book for courses on linear algebra, abstract algebra, and number theory at the upper-undergraduate level. It is also a valuable reference for researchers working in different fields of mathematics, computer science, and engineering as well as for individuals preparing for a career in mathematics education. |
| algebra and number theory: Algebraic Number Theory Ian Stewart, 1979-05-31 The title of this book may be read in two ways. One is 'algebraic number-theory', that is, the theory of numbers viewed algebraically; the other, 'algebraic-number theory', the study of algebraic numbers. Both readings are compatible with our aims, and both are perhaps misleading. Misleading, because a proper coverage of either topic would require more space than is available, and demand more of the reader than we wish to; compatible, because our aim is to illustrate how some of the basic notions of the theory of algebraic numbers may be applied to problems in number theory. Algebra is an easy subject to compartmentalize, with topics such as 'groups', 'rings' or 'modules' being taught in comparative isolation. Many students view it this way. While it would be easy to exaggerate this tendency, it is not an especially desirable one. The leading mathematicians of the nineteenth and early twentieth centuries developed and used most of the basic results and techniques of linear algebra for perhaps a hundred years, without ever defining an abstract vector space: nor is there anything to suggest that they suf fered thereby. This historical fact may indicate that abstrac tion is not always as necessary as one commonly imagines; on the other hand the axiomatization of mathematics has led to enormous organizational and conceptual gains. |
| algebra and number theory: Algebraic Number Theory Serge Lang, 2013-06-29 This is a second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Lang's books are always of great value for the graduate student and the research mathematician. This updated edition of Algebraic number theory is no exception.—-MATHEMATICAL REVIEWS |
| algebra and number theory: Algebraic Number Theory Jürgen Neukirch, 2013-03-14 This introduction to algebraic number theory discusses the classical concepts from the viewpoint of Arakelov theory. The treatment of class theory is particularly rich in illustrating complements, offering hints for further study, and providing concrete examples. It is the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available. |
| algebra and number theory: Lectures on the Theory of Algebraic Numbers E. T. Hecke, 2013-03-09 . . . 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. |
| algebra and number theory: Algebraic Number Theory A. Fröhlich, Martin J. Taylor, 1991 This book originates from graduate courses given in Cambridge and London. It provides a brisk, thorough treatment of the foundations of algebraic number theory, and builds on that to introduce more advanced ideas. Throughout, the authors emphasise the systematic development of techniques for the explicit calculation of the basic invariants, such as rings of integers, class groups, and units. Moreover they combine, at each stage of development, theory with explicit computations and applications, and provide motivation in terms of classical number-theoretic problems. A number of special topics are included that can be treated at this level but can usually only be found in research monographs or original papers, for instance: module theory of Dedekind domains; tame and wild ramifications; Gauss series and Gauss periods; binary quadratic forms; and Brauer relations. This is the only textbook at this level which combines clean, modern algebraic techniques together with a substantial arithmetic content. It will be indispensable for all practising and would-be algebraic number theorists. |
| algebra and number theory: A Course in Computational Algebraic Number Theory Henri Cohen, 2013-04-17 A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject. |
| algebra and number theory: Algebraic Number Theory Frazer Jarvis, 2014-06-23 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. |
| algebra and number theory: Algebraic Number Theory H. Koch, 2012-12-06 From the reviews: ... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes. Monatshefte fuer Mathematik, 1994 ... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory. Medelingen van het wiskundig genootschap, 1995 |
| algebra and number theory: Equations and Inequalities Jiri Herman, Radan Kucera, Jaromir Simsa, 2012-12-06 A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises. |
| algebra and number theory: A Brief Guide to Algebraic Number Theory H. P. F. Swinnerton-Dyer, 2001-02-22 Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author. |
| algebra and number theory: TwentyTwo Papers on Algebra, Number Theory and Differential Geometry M. S. Calenko, 1964-12-31 |
| algebra and number theory: Commutative Algebra David Eisenbud, 2013-12-01 This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included. |
| algebra and number theory: Fundamentals of Number Theory William J. LeVeque, 2014-01-05 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. |
| algebra and number theory: Algebraic Number Theory John Coates, 1989 |
| algebra and number theory: Algebraic Theory of Numbers Pierre Samuel, 2008 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. |
| algebra and number theory: Geometric Methods in Algebra and Number Theory Fedor Bogomolov, Yuri Tschinkel, 2006-06-22 * Contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory * The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry * Text can serve as an intense introduction for graduate students and those wishing to pursue research in algebraic and arithmetic geometry |
| algebra and number theory: Algebraic Number Theory Robert L. Long, 1977 |
| algebra and number theory: Computer Algebra and Polynomials Jaime Gutierrez, Josef Schicho, Martin Weimann, 2015-01-20 Algebra and number theory have always been counted among the most beautiful mathematical areas with deep proofs and elegant results. However, for a long time they were not considered that important in view of the lack of real-life applications. This has dramatically changed: nowadays we find applications of algebra and number theory frequently in our daily life. This book focuses on the theory and algorithms for polynomials over various coefficient domains such as a finite field or ring. The operations on polynomials in the focus are factorization, composition and decomposition, basis computation for modules, etc. Algorithms for such operations on polynomials have always been a central interest in computer algebra, as it combines formal (the variables) and algebraic or numeric (the coefficients) aspects. The papers presented were selected from the Workshop on Computer Algebra and Polynomials, which was held in Linz at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) during November 25-29, 2013, at the occasion of the Special Semester on Applications of Algebra and Number Theory. |
| algebra and number theory: An Introduction to Algebraic Number Theory Takashi Ono, 2012-12-06 This book is a translation of my book Suron Josetsu (An Introduction to Number Theory), Second Edition, published by Shokabo, Tokyo, in 1988. The translation is faithful to the original globally but, taking advantage of my being the translator of my own book, I felt completely free to reform or deform the original locally everywhere. When I sent T. Tamagawa a copy of the First Edition of the original work two years ago, he immediately pointed out that I had skipped the discussion of the class numbers of real quadratic fields in terms of continued fractions and (in a letter dated 2/15/87) sketched his idea of treating continued fractions without writing explicitly continued fractions, an approach he had first presented in his number theory lectures at Yale some years ago. Although I did not follow his approach exactly, I added to this translation a section (Section 4. 9), which nevertheless fills the gap pointed out by Tamagawa. With this addition, the present book covers at least T. Takagi's Shoto Seisuron Kogi (Lectures on Elementary Number Theory), First Edition (Kyoritsu, 1931), which, in turn, covered at least Dirichlet's Vorlesungen. It is customary to assume basic concepts of algebra (up to, say, Galois theory) in writing a textbook of algebraic number theory. But I feel a little strange if I assume Galois theory and prove Gauss quadratic reciprocity. |
| algebra and number theory: Number Theory Róbert Freud, Edit Gyarmati, 2020-10-08 Number Theory is a newly translated and revised edition of the most popular introductory textbook on the subject in Hungary. The book covers the usual topics of introductory number theory: divisibility, primes, Diophantine equations, arithmetic functions, and so on. It also introduces several more advanced topics including congruences of higher degree, algebraic number theory, combinatorial number theory, primality testing, and cryptography. The development is carefully laid out with ample illustrative examples and a treasure trove of beautiful and challenging problems. The exposition is both clear and precise. The book is suitable for both graduate and undergraduate courses with enough material to fill two or more semesters and could be used as a source for independent study and capstone projects. Freud and Gyarmati are well-known mathematicians and mathematical educators in Hungary, and the Hungarian version of this book is legendary there. The authors' personal pedagogical style as a facet of the rich Hungarian tradition shines clearly through. It will inspire and exhilarate readers. |
| algebra and number theory: A Classical Introduction to Modern Number Theory K. Ireland, M. Rosen, 2013-03-09 This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time. |
| algebra and number theory: Computational Algebra and Number Theory Wieb Bosma, Alf van der Poorten, 2013-03-09 Computers have stretched the limits of what is possible in mathematics. More: they have given rise to new fields of mathematical study; the analysis of new and traditional algorithms, the creation of new paradigms for implementing computational methods, the viewing of old techniques from a concrete algorithmic vantage point, to name but a few. Computational Algebra and Number Theory lies at the lively intersection of computer science and mathematics. It highlights the surprising width and depth of the field through examples drawn from current activity, ranging from category theory, graph theory and combinatorics, to more classical computational areas, such as group theory and number theory. Many of the papers in the book provide a survey of their topic, as well as a description of present research. Throughout the variety of mathematical and computational fields represented, the emphasis is placed on the common principles and the methods employed. Audience: Students, experts, and those performing current research in any of the topics mentioned above. |
| algebra and number theory: An Adventurer's Guide to Number Theory Richard Friedberg, 2012-07-06 This witty introduction to number theory deals with the properties of numbers and numbers as abstract concepts. Topics include primes, divisibility, quadratic forms, and related theorems. |
| algebra and number theory: Algebraic Number Theory Richard A. Mollin, 2011-01-05 Bringing the material up to date to reflect modern applications, this second edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. It offers a more complete and involved treatment of Galois theory, a more comprehensive section on Pollard's cubic factoring algorithm, and more detailed explanations of proofs to provide a sound understanding of challenging material. This edition also studies binary quadratic forms and compares the ideal and form class groups. The text includes convenient cross-referencing, a comprehensive index, and numerous exercises and applications. |
| algebra and number theory: Undergraduate Commutative Algebra Miles Reid, 1995-11-30 Commutative algebra is at the crossroads of algebra, number theory and algebraic geometry. This textbook is affordable and clearly illustrated, and is intended for advanced undergraduate or beginning graduate students with some previous experience of rings and fields. Alongside standard algebraic notions such as generators of modules and the ascending chain condition, the book develops in detail the geometric view of a commutative ring as the ring of functions on a space. The starting point is the Nullstellensatz, which provides a close link between the geometry of a variety V and the algebra of its coordinate ring A=k[V]; however, many of the geometric ideas arising from varieties apply also to fairly general rings. The final chapter relates the material of the book to more advanced topics in commutative algebra and algebraic geometry. It includes an account of some famous 'pathological' examples of Akizuki and Nagata, and a brief but thought-provoking essay on the changing position of abstract algebra in today's world. |
| algebra and number theory: Introduction to Cyclotomic Fields Lawrence C. Washington, 2012-12-06 This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat’s Last Theorem, and Iwasawa’s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott’s proof of the vanishing of Iwasawa’s f-invariant. |
| algebra and number theory: A Course in Number Theory H. E. Rose, 1995 This textbook covers the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. This book excludes topics relying heavily on complex analysis and advanced algebraic number theory. The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on Fermat's last theorem is also briefly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables. |
| algebra and number theory: Quadratic Number Theory J. L. Lehman, 2019-02-13 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. |
| algebra and number theory: The Theory of Algebraic Number Fields David Hilbert, 2013-03-14 A translation of Hilberts Theorie der algebraischen Zahlkörper best known as the Zahlbericht, first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher. |
| algebra and number theory: 111 Problems in Algebra and Number Theory Adrian Andreescu, Vinjai Vale, 2016 Algebra plays a fundamental role not only in mathematics, but also in various other scientific fields. Without algebra there would be no uniform language to express concepts such as numbers' properties. Thus one must be well-versed in this domain in order to improve in other mathematical disciplines. We cover algebra as its own branch of mathematics and discuss important techniques that are also applicable in many Olympiad problems. Number theory too relies heavily on algebraic machinery. Often times, the solutions to number theory problems involve several steps. Such a solution typically consists of solving smaller problems originating from a hypothesis and ending with a concrete statement that is directly equivalent to or implies the desired condition. In this book, we introduce a solid foundation in elementary number theory, focusing mainly on the strategies which come up frequently in junior-level Olympiad problems. |
| algebra and number theory: Algebraic Number Theory H. Koch, Helmut Koch, 1997-09-12 From the reviews of the first printing, published as Volume 62 of the Encyclopaedia of Mathematical Sciences: ... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes. Monatshefte fuer Mathematik, 1994 ... Koch's book is written mostly for non-specialists. It is an up-to-date account of the subject dealing with mostly general questions. Special results appear only as illustrating examples for the general features of the theory. It is supposed that the reader has good general background in the fields of modern (abstract) algebra and elementary number theory. We recommend this volume mainly to graduate studens and research mathematicians. Acta Scientiarum Mathematicarum, 1993 |
| algebra and number theory: Number Theory W.A. Coppel, 2006-02-02 This two-volume book is a modern introduction to the theory of numbers, emphasizing its connections with other branches of mathematics. Part A is accessible to first-year undergraduates and deals with elementary number theory. Part B is more advanced and gives the reader an idea of the scope of mathematics today. The connecting theme is the theory of numbers. By exploring its many connections with other branches a broad picture is obtained. The book contains a treasury of proofs, several of which are gems seldom seen in number theory books. |
| algebra and number theory: Primes of the Form X2 + Ny2 David A. Cox, 1989-09-28 Modern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of field theory and its intimate connection with complex multiplication. While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included. The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively. |
Algebra - Wikipedia
Elementary algebra, also called school algebra, college algebra, and classical algebra, [22] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on …
Introduction to Algebra - Math is Fun
Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6". But instead of saying " obviously x=6", use this neat step-by-step …
Algebra I - Khan Academy
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a …
Algebra | History, Definition, & Facts | Britannica
May 9, 2025 · Algebra is the branch of mathematics in which abstract symbols, rather than numbers, are manipulated or operated with arithmetic. For example, x + y = z or b - 2 = 5 are …
Algebra - What is Algebra? | Basic Algebra | Definition - Cuemath
Algebra is the branch of mathematics that represents problems in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, …
How to Understand Algebra (with Pictures) - wikiHow
Mar 18, 2025 · Algebra is a system of manipulating numbers and operations to try to solve problems. When you learn algebra, you will learn the rules to follow for solving problems. But …
What is Algebra? - BYJU'S
Algebra is one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the …
Algebra in Math - Definition, Branches, Basics and Examples
Apr 7, 2025 · This section covers key algebra concepts, including expressions, equations, operations, and methods for solving linear and quadratic equations, along with polynomials …
Algebra - Simple English Wikipedia, the free encyclopedia
People who do algebra use the rules of numbers and mathematical operations used on numbers. The simplest are adding, subtracting, multiplying, and dividing. More advanced operations …
OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Free algebra tutorial and help. Notes, videos, steps. Solve and simplify linear, quadratic, polynomial, and rational expressions and equations.
Algebra - Wikipedia
Elementary algebra, also called school algebra, college algebra, and classical algebra, [22] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on …
Introduction to Algebra - Math is Fun
Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6". But instead of saying " obviously x=6", use this neat step-by-step …
Algebra I - Khan Academy
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a …
Algebra | History, Definition, & Facts | Britannica
May 9, 2025 · Algebra is the branch of mathematics in which abstract symbols, rather than numbers, are manipulated or operated with arithmetic. For example, x + y = z or b - 2 = 5 are …
Algebra - What is Algebra? | Basic Algebra | Definition - Cuemath
Algebra is the branch of mathematics that represents problems in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, …
How to Understand Algebra (with Pictures) - wikiHow
Mar 18, 2025 · Algebra is a system of manipulating numbers and operations to try to solve problems. When you learn algebra, you will learn the rules to follow for solving problems. But …
What is Algebra? - BYJU'S
Algebra is one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the …
Algebra in Math - Definition, Branches, Basics and Examples
Apr 7, 2025 · This section covers key algebra concepts, including expressions, equations, operations, and methods for solving linear and quadratic equations, along with polynomials …
Algebra - Simple English Wikipedia, the free encyclopedia
People who do algebra use the rules of numbers and mathematical operations used on numbers. The simplest are adding, subtracting, multiplying, and dividing. More advanced operations …
OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Free algebra tutorial and help. Notes, videos, steps. Solve and simplify linear, quadratic, polynomial, and rational expressions and equations.
