# BEACHY AND BLAIR ABSTRACT ALGEBRA PDF

A number theory thread runs throughout several optional sections, and there is an overview of techniques for computing Galois groups. The book offers an extensive set of exercises that help to build skills in writing proofs. Chapter introductions, together with notes at the ends of certain chapters, provide motivation and historical context, while relating the subject matter to the broader mathematical picture. Although the book starts in a very concrete fashion, we increase the level of sophistication as the book progresses, and, by the end of Chapter 6, all of the topics taught in our two semester sequence have been covered. It is our conviction that the level of sophistication should increase, slowly at first, as the students become familiar with the subject. We think our ordering of the topics speaks directly to this assertion.

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A number theory thread runs throughout several optional sections, and there is an overview of techniques for computing Galois groups. The book offers an extensive set of exercises that help to build skills in writing proofs.

Chapter introductions provide motivation and historical context, while relating the subject matter to the broader mathematical picture. FEATURES Progresses students from writing proofs in the familiar setting of the integers to dealing with abstract concepts once they have gained some confidence.

Separating the two hurdles of devising proofs and grasping abstract mathematics makes abstract algebra more accessible. Makes a concerted effort throughout to develop key examples in detail before introducing the relevant abstract definitions. For example, cyclic groups are introduced in Chapter 1 in the context of number theory, and permutations are studied in Chapter 2, before abstract groups are introduced in Chapter 3.

The ring of integers and rings of polynomials are covered before abstract rings are introduced in Chapter 5. Provides chapter introductions that give motivation and historical context while tying the subject matter in with the broader picture.

The text emphasizes the historical connections to the solution of polynomial equations and to the theory of numbers. For strong classes, there is a complete treatment of Galois theory, and for honors students, there are optional sections on advanced number theory topics. Recognizes the developing maturity of students by raising the writing level as the book progresses. The first two chapters on the integers and functions contain full details, in addition to comments on techniques of proof.

The intermediate chapters on groups, rings, and fields are written at a standard undergraduate level. Includes such optional topics as finite fields, the Sylow theorems, finite abelian groups, the simplicity of PSL 2,F , Euclidean domains, unique factorization domains, cyclotomic polynomials, arithmetic functions, Moebius inversion, quadratic reciprocity, primitive roots, and diophantine equations.

Offers an extensive set of exercises that provides ample opportunity for students to develop their ability to write proofs.

It contains solutions to all exercises. Our book is intended for this course, and has grown directly out of our experience in teaching the course at Northern Illinois University.

As a prerequisite to the abstract algebra course, our students are required to have taken a sophomore level course in linear algebra that is largely computational, although they have been introduced to proofs to some extent. Our classes include students preparing to teach high school, but almost no computer science or engineering students.

We certainly do not assume that all of our students will go on to graduate school in pure mathematics. In searching for appropriate text books, we have found several texts that start at about the same level as we do, but most of these stay at that level, and they do not teach nearly as much mathematics as we desire. On the other hand, there are several fine books that start and finish at the level of our Chapters 3 through 6, but these books tend to begin immediately with the abstract notion of group or ring , and then leave the average student at the starting gate.

We have in the past used such books, supplemented by our Chapter 1. Although the book starts in a very concrete fashion, we increase the level of sophistication as the book progresses, and, by the end of Chapter 6, all of the topics taught in our course have been covered.

It is our conviction that the level of sophistication should increase, slowly at first, as the students become familiar with the subject. We think our ordering of the topics speaks directly to this assertion. It is our feeling that such inclusions often tend to be superficial.

In order to make room for the inclusion of applications, some important mathematical concepts have to be sacrificed. It is clear that one must have substantial experience with abstract algebra before any genuine applications can be treated.

For this reason we feel that the most honest introduction concentrates on the algebra. One of the reasons frequently given for treating applications is that they motivate the student. We prefer to motivate the subject with concrete problems from areas that the students have previously encountered, namely, the integers and polynomials over the real numbers. One problem with most treatments of abstract algebra, whether they begin with group theory or ring theory, is that the students simultaneously encounter for the first time both abstract mathematics and the requirement that they produce proofs of their own devising.

By taking a more concrete approach than is usual, we hope to separate these two initiations. In three of the first four chapters of our book we discuss familiar concrete mathematics: number theory, functions and permutations, and polynomials.

Although the objects of study are concrete, and most are familiar, we cover quite a few nontrivial ideas and at the same time introduce the student to the subtle ideas of mathematical proof. At Northern Illinois University, this course and Advanced Calculus are the traditional places for students to learn how to write proofs. After studying Chapters 1 and 2, the students have at their disposal some of the most important examples of groups-permutation groups, the group of integers modulo n, and certain matrix groups.

In Chapter 3 the abstract definition of a group is introduced, and the students encounter the notion of a group armed with a variety of concrete examples. Probably the most difficult notion in elementary group theory is that of a factor group. Again this is a case where the difficulty arises because there are, in fact, two new ideas encountered together.

We have tried to separate these by treating the notions of equivalence relation and partition in Chapter 2 in the context of sets and functions. These ideas are related to the integers modulo n, studied in Chapter 1. When factor groups are introduced in Chapter 3, we have partitions and equivalence relations at our disposal, and we are able to concentrate on the group structure introduced on the equivalence classes.

In Chapter 4 we return to a more concrete subject when we derive some important properties of polynomials. Chapter 5 then introduces the abstract definition of a ring after we have already encountered several important examples of rings: the integers, the integers modulo n, and the ring of polynomials with coefficients in any field.

From this point on our book looks more like a traditional abstract algebra textbook. After rings we consider fields, and we include a discussion of root adjunction as well as the three problems from antiquity: squaring the circle, duplicating the cube, and trisecting an angle. We also discuss splitting fields and finite fields here. We feel that the first six chapters represent the most that students at institutions such as ours can reasonably absorb in a year.

Chapter 7 returns to group theory to consider several more sophisticated ideas including those needed for Galois theory, which is the subject matter of Chapter 8. In Chapter 9 we return to a study of rings, and consider questions of unique factorization. In fact, this is the last of a thread of number theoretic applications that run through the text, including a proof of the quadratic reciprocity law in Section 6.

The applications to number theory provide topics suitable for honors students. The last three chapters are intended to make the book suitable for an honors course or for classes of especially talented or well-prepared students.

In these chapters the writing style is rather terse and demanding. The only prerequisite for our text is a sophomore level course in linear algebra.

We do not assume that the student has been required to write, or even read, proofs before taking our course. We do use examples from matrix algebra in our discussion of group theory, and we draw on the computational techniques learned in the linear algebra course-see, for example, our treatment of the Euclidean algorithm in Chapter 1.

We have included a number of appendices to which the student may be referred for background material. The appendices on induction and on the complex numbers might be appropriate to cover in class, and so they include some exercises. In our classes we usually intend to cover Chapters 1, 2 and 3 in the first semester, and most of Chapters 4, 5 and 6 in the second semester. In practice, we usually begin the second semester with group homomorphisms and factor groups, and end with geometric constructions.

We have rarely had time to cover splitting fields and finite fields. For students with better preparation, Chapters 1 and 2 could be covered more quickly. The development is arranged so that Chapter 7 on the structure of groups can be covered immediately after Chapter 3.

On the other hand, the material from Chapter 7 is not really needed until Section 8. We have included answers to some of the odd numbered computational exercises. Back to the top of the page.

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Faunos Vincent Russo rated it really liked it Sep 06, Highly regarded by instructors in past editions for its sequencing of topics as well as its concrete approach, slightly slower beginning pace, and extensive set of exercises, the latest edition of Abstract Algebra extends the thrust of the widely used beacuy editions as it introduces modern abstract concepts only after a careful study of important examples. We have also benefitted over the years from numerous comments from our own students and from abstrsct variety of colleagues. Kiana Kaviany marked it as to-read Oct 12, A number theory thread runs throughout several optional sections, and there is an overview of techniques for computing Galois groups. Martin Cohen marked it as to-read Apr 22, I like this balance very much. Swetha marked it as to-read Nov 19, Blaur m I would first like to point out that this is bewchy slow pace subject.

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## ABSTRACT ALGEBRA BEACHY BLAIR PDF

A number theory thread runs throughout several optional sections, and there is an overview of techniques for computing Galois groups. The book offers an extensive set of exercises that help to build skills in writing proofs. Chapter introductions provide motivation and historical context, while relating the subject matter to the broader mathematical picture. FEATURES Progresses students from writing proofs in the familiar setting of the integers to dealing with abstract concepts once they have gained some confidence. Separating the two hurdles of devising proofs and grasping abstract mathematics makes abstract algebra more accessible. Makes a concerted effort throughout to develop key examples in detail before introducing the relevant abstract definitions. For example, cyclic groups are introduced in Chapter 1 in the context of number theory, and permutations are studied in Chapter 2, before abstract groups are introduced in Chapter 3.

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## Abstract Algebra

Gosida The ring of integers and rings of polynomials are covered before abstract rings are introduced in Chapter 5. I had to read it for my first algebra course, and I hated it. Rather than outlining a large number of possible paths through various parts of the text, we have to ask the instructor to read ahead and use a great deal of caution in choosing any paths other than the ones we have suggested above. Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book.

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## by J.A.Beachy and W.D.Blair

After using the book, on more than one occasion he sent us a large number of detailed suggestions on how to improve the presentation. Many of these were in response to questions from his students, so we owe an enormous debt of gratitude to his students, as well as to Professor Bergman. We believe that our responses to his suggestions and corrections have measurably improved the book. We would also like to acknowledge important corrections and suggestions that we received from Marie Vitulli, of the University of Oregon, and from David Doster, of Choate Rosemary Hall. We have also benefitted over the years from numerous comments from our own students and from a variety of colleagues. We would like to add Doug Bowman, Dave Rusin, and Jeff Thunder to the list of colleagues given in the preface to the second edition. In this edition we have added about exercises, we have added 1 to all rings, and we have done our best to weed out various errors and misprints.