**Author**: Steven J. Rosenberg

**Publisher:**CRC Press

**ISBN:**1000516318

**Category :**Mathematics

**Languages :**en

**Pages :**390

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## An Invitation to Abstract Algebra

**Author**: Steven J. Rosenberg

**Publisher:** CRC Press

**ISBN:** 1000516318

**Category : **Mathematics

**Languages : **en

**Pages : **390

**Book Description**

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout. The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can—and cannot—represent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks—period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

## An Invitation to Abstract Algebra

**Author**: Steven J. Rosenberg

**Publisher:** CRC Press

**ISBN:** 1000516318

**Category : **Mathematics

**Languages : **en

**Pages : **390

**Book Description**

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout. The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can—and cannot—represent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks—period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

## An Invitation to Abstract Algebra

**Author**: STEVEN J. ROSENBERG

**Publisher:** CRC Press

**ISBN:** 9780367748616

**Category : **

**Languages : **en

**Pages : **372

**Book Description**

Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks. This book is offered as a manual to a new way of thinking. The author aims to instill the desire to understand the material, to encourage more discovery, and to appreciate the subject for its own sake.

## An Invitation to Abstract Mathematics

**Author**: Béla Bajnok

**Publisher:** Springer Nature

**ISBN:** 3030561747

**Category : **Mathematics

**Languages : **en

**Pages : **442

**Book Description**

This undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics. Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise. This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts. From reviews of the first edition: Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. ... The book can be used as a text for traditional transition or structure courses ... but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA Reviews The style of writing is careful, but joyously enthusiastic.... The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background ... and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATH

## Abstract Algebra

**Author**: Ronald Solomon

**Publisher:** American Mathematical Soc.

**ISBN:** 9780821847954

**Category : **Mathematics

**Languages : **en

**Pages : **227

**Book Description**

This undergraduate text takes a novel approach to the standard introductory material on groups, rings, and fields. At the heart of the text is a semi-historical journey through the early decades of the subject as it emerged in the revolutionary work of Euler, Lagrange, Gauss, and Galois. Avoiding excessive abstraction whenever possible, the text focuses on the central problem of studying the solutions of polynomial equations. Highlights include a proof of the Fundamental Theorem of Algebra, essentially due to Euler, and a proof of the constructability of the regular 17-gon, in the manner of Gauss. Another novel feature is the introduction of groups through a meditation on the meaning of congruence in the work of Euclid. Everywhere in the text, the goal is to make clear the links connecting abstract algebra to Euclidean geometry, high school algebra, and trigonometry, in the hope that students pursuing a career as secondary mathematics educators will carry away a deeper and richer understanding of the high school mathematics curriculum. Another goal is to encourage students, insofar as possible in a textbook format, to build the course for themselves, with exercises integrally embedded in the text of each chapter.

## An Invitation to Algebraic Geometry

**Author**: Karen Smith

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387989808

**Category : **Mathematics

**Languages : **en

**Pages : **164

**Book Description**

This is a description of the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra.

## Abstract Algebra with Applications

**Author**: Audrey Terras

**Publisher:** Cambridge University Press

**ISBN:** 1107164079

**Category : **Mathematics

**Languages : **en

**Pages : **328

**Book Description**

Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world. The first part of this book covers groups, after some preliminaries on sets, functions, relations, and induction, and features applications such as public-key cryptography, Sudoku, the finite Fourier transform, and symmetry in chemistry and physics. The second part of this book covers rings and fields, and features applications such as random number generators, error correcting codes, the Google page rank algorithm, communication networks, and elliptic curve cryptography. The book's masterful use of colorful figures and images helps illustrate the applications and concepts in the text. Real-world examples and exercises will help students contextualize the information. Meant for a year-long undergraduate course in algebra for mathematics, engineering, and computer science majors, the only prerequisites are calculus and a bit of courage when asked to do a short proof.

## Abstract Linear Algebra

**Author**: Morton L. Curtis

**Publisher:** Springer Science & Business Media

**ISBN:** 1441987649

**Category : **Mathematics

**Languages : **en

**Pages : **168

**Book Description**

Intended for a first course on the subject, this text begins from scratch and develops the standard topics of Linear Algebra. Its progresses simply towards its ultimate goal, the Theorem of Hurwitz, which argues that the only normed algebras over the real numbers are the real numbers, the complex numbers, the quaternions, and the octonions. The book stresses the complete logical development of the subject.

## An Invitation to C*-Algebras

**Author**: W. Arveson

**Publisher:** Springer Science & Business Media

**ISBN:** 1461263719

**Category : **Mathematics

**Languages : **en

**Pages : **108

**Book Description**

This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-algebras become GCR. This practice probably creates an impression that nothing of value is known about other C*-algebras. Of course that is not true. But insofar as representations are con cerned, we can point to the empirical fact that to this day no one has given a concrete parametric description of even the irreducible representations of any C*-algebra which is not GCR. Indeed, there is metamathematical evidence which strongly suggests that no one ever will (see the discussion at the end of Section 3. 4). Occasionally, when the idea behind the proof of a general theorem is exposed very clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. 1 and 2. 2. (it would be desirable but not necessary to include Section 1. 1 as well), whereas someone interested in Borel structures could read Chapter 3 separately. Chapter I could be used as a bare-bones introduction to C*-algebras. Sections 2.

## CSU Abstract Algebra

**Author**:

**Publisher:** PediaPress

**ISBN:**

**Category : **

**Languages : **en

**Pages : **

**Book Description**

## Abstract Algebra

**Author**: Pierre Antoine Grillet

**Publisher:** Springer Science & Business Media

**ISBN:** 0387715681

**Category : **Mathematics

**Languages : **en

**Pages : **674

**Book Description**

A completely reworked new edition of this superb textbook. This key work is geared to the needs of the graduate student. It covers, with proofs, the usual major branches of groups, rings, fields, and modules. Its inclusive approach means that all of the necessary areas are explored, while the level of detail is ideal for the intended readership. The text tries to promote the conceptual understanding of algebra as a whole, doing so with a masterful grasp of methodology. Despite the abstract subject matter, the author includes a careful selection of important examples, together with a detailed elaboration of the more sophisticated, abstract theories.

eBook Journalism in PDF, ePub, Mobi and Kindle

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout. The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can—and cannot—represent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks—period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout. The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can—and cannot—represent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks—period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks. This book is offered as a manual to a new way of thinking. The author aims to instill the desire to understand the material, to encourage more discovery, and to appreciate the subject for its own sake.

This undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics. Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise. This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts. From reviews of the first edition: Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. ... The book can be used as a text for traditional transition or structure courses ... but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA Reviews The style of writing is careful, but joyously enthusiastic.... The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background ... and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATH

This undergraduate text takes a novel approach to the standard introductory material on groups, rings, and fields. At the heart of the text is a semi-historical journey through the early decades of the subject as it emerged in the revolutionary work of Euler, Lagrange, Gauss, and Galois. Avoiding excessive abstraction whenever possible, the text focuses on the central problem of studying the solutions of polynomial equations. Highlights include a proof of the Fundamental Theorem of Algebra, essentially due to Euler, and a proof of the constructability of the regular 17-gon, in the manner of Gauss. Another novel feature is the introduction of groups through a meditation on the meaning of congruence in the work of Euclid. Everywhere in the text, the goal is to make clear the links connecting abstract algebra to Euclidean geometry, high school algebra, and trigonometry, in the hope that students pursuing a career as secondary mathematics educators will carry away a deeper and richer understanding of the high school mathematics curriculum. Another goal is to encourage students, insofar as possible in a textbook format, to build the course for themselves, with exercises integrally embedded in the text of each chapter.

This is a description of the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra.

Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world. The first part of this book covers groups, after some preliminaries on sets, functions, relations, and induction, and features applications such as public-key cryptography, Sudoku, the finite Fourier transform, and symmetry in chemistry and physics. The second part of this book covers rings and fields, and features applications such as random number generators, error correcting codes, the Google page rank algorithm, communication networks, and elliptic curve cryptography. The book's masterful use of colorful figures and images helps illustrate the applications and concepts in the text. Real-world examples and exercises will help students contextualize the information. Meant for a year-long undergraduate course in algebra for mathematics, engineering, and computer science majors, the only prerequisites are calculus and a bit of courage when asked to do a short proof.

Intended for a first course on the subject, this text begins from scratch and develops the standard topics of Linear Algebra. Its progresses simply towards its ultimate goal, the Theorem of Hurwitz, which argues that the only normed algebras over the real numbers are the real numbers, the complex numbers, the quaternions, and the octonions. The book stresses the complete logical development of the subject.

This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-algebras become GCR. This practice probably creates an impression that nothing of value is known about other C*-algebras. Of course that is not true. But insofar as representations are con cerned, we can point to the empirical fact that to this day no one has given a concrete parametric description of even the irreducible representations of any C*-algebra which is not GCR. Indeed, there is metamathematical evidence which strongly suggests that no one ever will (see the discussion at the end of Section 3. 4). Occasionally, when the idea behind the proof of a general theorem is exposed very clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. 1 and 2. 2. (it would be desirable but not necessary to include Section 1. 1 as well), whereas someone interested in Borel structures could read Chapter 3 separately. Chapter I could be used as a bare-bones introduction to C*-algebras. Sections 2.

A completely reworked new edition of this superb textbook. This key work is geared to the needs of the graduate student. It covers, with proofs, the usual major branches of groups, rings, fields, and modules. Its inclusive approach means that all of the necessary areas are explored, while the level of detail is ideal for the intended readership. The text tries to promote the conceptual understanding of algebra as a whole, doing so with a masterful grasp of methodology. Despite the abstract subject matter, the author includes a careful selection of important examples, together with a detailed elaboration of the more sophisticated, abstract theories.