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Introduction to Abstract Algebra
Abstract algebra is an essential tool inalgebra, number theory, geometry, topology, and, to a lesser extent, analysis. Thisbook is intended as a textbook for a one-term senior undergraduate or gradatecourse in abstract algebra to prepare students for further readings on relevantsubjects such as Group Theory and Galois Theory.
eBook
Introduction to Modern Algebra and its Applications
\"The book provides an introduction to modern abstract algebra and its applications. It covers all major topics of classical theory of numbers, groups, rings, fields and finite dimensional algebras. The book also provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. In particular, it considers algorithm RSA, secret sharing algorithms, Diffie-Hellman Scheme and ElGamal cryptosystem based on discrete logarithm problem. It also presents Buchbergers algorithm which is one of the important algorithms for constructing Gröbner basis. Key Features: Covers all major topics of classical theory of modern abstract algebra such as groups, rings and fields and their applications. In addition it provides the introduction to the number theory, theory of finite fields, finite dimensional algebras and their applications. Provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. Presents numerous examples illustrating the theory and applications. It is also filled with a number of exercises of various difficulty.
Describes in detail the construction of the Cayley-Dickson construction for finite dimensional algebras, in particular, algebras of quaternions and octonions and gives their applications in the number theory and computer graphics.\"
eBook
Polyadic algebraic structures
Various algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary (\"polyadization\" and partial arity freedom principle), and the relations between operations, arising from the structure definitions, lead to restrictions of their arity shapes (\"quantization\"). This unified procedure is applied to one-set and two-set algebraic structures, as well as to Hopf algebras and tensor categories, which gives new unusual properties and enriched objects absent in the ordinary binary structures.
eBook
Polygroup theory and related systems
This monograph is devoted to the study of Polygroup Theory. It begins with some basic results concerning group theory and algebraic hyperstructures, which represent the most general algebraic context, in which reality can be modeled. Most results on polygroups are collected in this book. Moreover, this monograph is the first book on this theory. The volume is highly recommended to theoreticians in pure and applied mathematics.
eBook
A First Graduate Course in Abstract Algebra
Divided into two sections, this book covers both the standard topics (groups, modules, rings, and vector spaces) associated with abstract algebra and more advanced topics such as Galois fields, noncommutative rings, group extensions, and Abelian groups. The author includes review material where needed instead of in a single chapter, giving convenient access with minimal page turning. He also provides ample examples, exercises, and problem sets to reinforce the material. This book illustrates the theory of finitely generated modules over principal ideal domains, discusses tensor products, and demonstrates the development of determinants. It also covers Sylow theory and Jordan canonical form.
eBook
Rule Learning by Seven-Month-Old Infants
A fundamental task of language acquisition is to extract abstract algebraic rules. Three experiments show that 7-month-old infants attend longer to sentences with unfamiliar structures than to sentences with familiar structures. The design of the artificial language task used in these experiments ensured that this discrimination could not be performed by counting, by a system that is sensitive only to transitional probabilities, or by a popular class of simple neural network models. Instead, these results suggest that infants can represent, extract, and generalize abstract algebraic rules.
Journal Article
Hypergeometric functions over finite fields
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we
consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions
over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the
classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric
transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As
an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation
formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain
hypergeometric varieties.
eBook