See S&T, page 19, Theorem 1.8. De nition 1.3. A number eld (or algebraic number eld) is a nite ex-tension Kof Q. The index K: Q is the degree of the number eld. If Kis a number eld then K= Q( ) for some (algebraic) number 2K. See S&T, page 32, Theorem 2.2. Let K= Q( ) be a number eld of degree nover Q.
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers, and p-adic integers.
- 1Research fields
Research fields[edit]
Active research areas[edit]
Basic notions[edit]
- Ring ideal, maximal ideal, prime ideal
- Ring homomorphism
Classes of rings[edit]
- Nilpotent elements and reduced rings
Constructions with commutative rings[edit]
Localization and completion[edit]
- Localization of a ring
- Valuation (mathematics)
Finiteness properties[edit]
- Ascending chain condition (ACC) and descending chain condition (DCC)
Ideal theory[edit]
Homological properties[edit]
Dimension theory[edit]
- Regular local ring
Ring extensions, primary decomposition[edit]
- Primary decomposition and the Lasker–Noether theorem
Relation with algebraic geometry[edit]
Computational and algorithmic aspects[edit]
Active research areas[edit]
Related disciplines[edit]
Retrieved from 'https://en.wikipedia.org/w/index.php?title=List_of_commutative_algebra_topics&oldid=743829796'
Download Book A Computational Introduction To Number Theory And Algebra in PDF format. You can Read Online A Computational Introduction To Number Theory And Algebra here in PDF, EPUB, Mobi or Docx formats.A Computational Introduction To Number Theory And Algebra
Author :Victor ShoupISBN :9780521516440
Genre :Computers
File Size : 27.1 MB
Format :PDF, ePub, Docs
Download :686
Read :754
An introductory graduate-level text emphasizing algorithms and applications. This second edition includes over 200 new exercises and examples.
Computational Number Theory And Modern Cryptography
Author :Song Y. YanISBN :9781118188613
Genre :Computers
File Size : 40.25 MB
Format :PDF, ePub
Download :154
Read :524
The only book to provide a unified view of the interplay betweencomputational number theory and cryptography Computational number theory and modern cryptography are two ofthe most important and fundamental research fields in informationsecurity. In this book, Song Y. Yang combines knowledge of thesetwo critical fields, providing a unified view of the relationshipsbetween computational number theory and cryptography. The authortakes an innovative approach, presenting mathematical ideas first,thereupon treating cryptography as an immediate application of themathematical concepts. The book also presents topics from numbertheory, which are relevant for applications in public-keycryptography, as well as modern topics, such as coding and latticebased cryptography for post-quantum cryptography. The authorfurther covers the current research and applications for commoncryptographic algorithms, describing the mathematical problemsbehind these applications in a manner accessible to computerscientists and engineers. Makes mathematical problems accessible to computer scientistsand engineers by showing their immediate application Presents topics from number theory relevant for public-keycryptography applications Covers modern topics such as coding and lattice basedcryptography for post-quantum cryptography Starts with the basics, then goes into applications and areasof active research Geared at a global audience; classroom tested in North America,Europe, and Asia Incudes exercises in every chapter Instructor resources available on the book’s CompanionWebsite Computational Number Theory and Modern Cryptography isideal for graduate and advanced undergraduate students incomputer science, communications engineering, cryptography andmathematics. Computer scientists, practicing cryptographers, andother professionals involved in various security schemes will alsofind this book to be a helpful reference.
A Course In Computational Algebraic Number Theory
Author :Henri CohenISBN :9783662029459
Genre :Mathematics
File Size : 56.56 MB
Format :PDF, ePub, Docs
Download :526
Read :1254
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.
Circuits And Systems For Security And Privacy
Author :Farhana SheikhISBN :9781482236897
Genre :Computers
File Size : 65.41 MB
Format :PDF, Docs
Download :216
Read :921
Circuits and Systems for Security and Privacy begins by introducing the basic theoretical concepts and arithmetic used in algorithms for security and cryptography, and by reviewing the fundamental building blocks of cryptographic systems. It then analyzes the advantages and disadvantages of real-world implementations that not only optimize power, area, and throughput but also resist side-channel attacks. Merging the perspectives of experts from industry and academia, the book provides valuable insight and necessary background for the design of security-aware circuits and systems as well as efficient accelerators used in security applications.
Handbook Of Finite Fields
Author :Gary L. MullenISBN :9781439873823
Genre :Computers
File Size : 52.7 MB
Format :PDF, Docs
Download :877
Read :345
Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and each chapter is self contained and peer reviewed. The first part of the book traces the history of finite fields through the eighteenth and nineteenth centuries. The second part presents theoretical properties of finite fields, covering polynomials, special functions, sequences, algorithms, curves, and related computational aspects. The final part describes various mathematical and practical applications of finite fields in combinatorics, algebraic coding theory, cryptographic systems, biology, quantum information theory, engineering, and other areas. The book provides a comprehensive index and easy access to over 3,000 references, enabling you to quickly locate up-to-date facts and results regarding finite fields.
Computational Complexity
Author :Sanjeev AroraISBN :1139477366
Genre :Computers
File Size : 46.85 MB
Format :PDF
Download :677
Read :316
This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set. The book starts with a broad introduction to the field and progresses to advanced results. Contents include: definition of Turing machines and basic time and space complexity classes, probabilistic algorithms, interactive proofs, cryptography, quantum computation, lower bounds for concrete computational models (decision trees, communication complexity, constant depth, algebraic and monotone circuits, proof complexity), average-case complexity and hardness amplification, derandomization and pseudorandom constructions, and the PCP theorem.
An Experimental Introduction To Number Theory
Author :Benjamin HutzISBN :9781470430979
Genre :Number theory
File Size : 48.54 MB
Format :PDF, Docs
Download :225
Read :243
This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) open-ended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems. The mathematical prerequisites for this book are few. Early chapters contain topics such as integer divisibility, modular arithmetic, and applications to cryptography, while later chapters contain more specialized topics, such as Diophantine approximation, number theory of dynamical systems, and number theory with polynomials. Students of all levels will be drawn in by the patterns and relationships of number theory uncovered through data driven exploration.
Mathematical Reviews
Author :ISBN :UOM:39015067268261
Genre :Mathematics
File Size : 61.10 MB
Format :PDF, Docs
Download :330
Read :1306
Advanced Linear Algebra
Author :Steven RomanISBN :9780387728315
Genre :Mathematics
File Size : 81.82 MB
Format :PDF, Mobi
Download :925
Read :1261
This graduate level textbook covers an especially broad range of topics. The book first offers a careful discussion of the basics of linear algebra. It then proceeds to a discussion of modules, emphasizing a comparison with vector spaces, and presents a thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory, culminating in the finite dimensional spectral theorem for normal operators. The new edition has been revised and contains a chapter on the QR decomposition, singular values and pseudoinverses, and a chapter on convexity, separation and positive solutions to linear systems.
Number Theory
Author :George E. AndrewsISBN :9780486135106
Genre :Mathematics
File Size : 34.5 MB
Format :PDF, Docs
Download :451
Read :1275
Undergraduate text uses combinatorial approach to accommodate both math majors and liberal arts students. Covers the basics of number theory, offers an outstanding introduction to partitions, plus chapters on multiplicativity-divisibility, quadratic congruences, additivity, and more