# ABSTRACT ALGEBRA PDF

ABSTRACT ALGEBRA. Second Edition. Charles C. Pinter. Professor of Mathematics. Bucknell University. Dover Publications, Inc., Mineola, New York. MA Introduction to Abstract Algebra. Samir Siksek. Mathematics Institute. University of Warwick. DIRE WARNING: These notes are printed on paper laced. I covered this material in a two-semester graduate course in abstract algebra in The main novelty is that most of the standard exercises in abstract algebra are.

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subject of abstract algebra and no student should go through such a However, one of the major problems in teaching an abstract algebra. then you might get something along the lines of: “Algebra is the abstract The central idea behind abstract algebra is to define a larger class of objects (sets. The purpose of this book is to complement the lectures and thereby decrease, but not eliminate, the necessity of taking lecture notes. Reading the appropri-.

David S. Dummit and Richard M. Foote are the authors of Abstract Algebra, 3rd Edition, published by Wiley. Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics.

I seldom used this book to teach myself a subject without first learning about it in lecture, so I can't comment too much here. The price is fairly high, but the book is huge. There is plenty of material here and it's arguably one of the few math books worth the list price.

Nevertheless, you'd be better off finding a used copy as there are most likely many people who could not handle the subject and returned the book. If you're serious about learning algebra this is a must-have book.

It will prepare you well for more advanced studies. There also seems to be a problem with the binding as my book and many of my classmates had problems with the cover ripping or spine splitting.

This is probably one of the best introductions to higher mathematics. The text provides a very clear foundation for much of the concepts encountered in the field, and gives numerous excellent examples of more refined and rigorous proofs. I would suggest perusing the work even before undergraduate education particularly if you have a clear interest in mathematics , as the mode of thought presented therein will certainly be a boon to your analytic endeavors at all levels.

## Abstract Algebra: The Basic Graduate Year

Group theory, after all, does not even require a complete mastery of advanced calculus or linear algebra to be understood acceptably, so the sooner you expose yourself to the concepts in Dummit and Foote, the better.

See all 98 reviews. Amazon Giveaway allows you to run promotional giveaways in order to create buzz, reward your audience, and attract new followers and customers. Learn more about Amazon Giveaway. This item: Abstract Algebra, 3rd Edition. Set up a giveaway. Customers who viewed this item also viewed. A Book of Abstract Algebra: Second Edition Dover Books on Mathematics. Charles C Pinter. Michael Artin. First Course in Abstract Algebra. John B. Pages with related products. See and discover other items: There's a problem loading this menu right now.

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Amazon Advertising Find, attract, and engage customers. Amazon Drive Cloud storage from Amazon. Alexa Actionable Analytics for the Web. AmazonGlobal Ship Orders Internationally. William G. Yandl, Seattle Univer- sity; and Lance L. Littlejohn, Utah State University. In the fifth edition: In the sixth edition: Daniel D. Golik, University of Missouri at St. In the seventh edition: Ali A. In the eighth edition: Ellington, Virginia Commonwealth University.

We thank Vera Pless, University of Illinois at Chicago, for critically read- ing the material on coding theory. We also wish to thank the following for their help with selected portions of the manuscript: Thomas I.

Bartlow, Robert E. Beck, and Michael L. Levitan, all of Villanova University; Robert C. Busby, Robin Clark, the late Charles S. Duris, Herman E. Gollwitzer, Milton Schwartz, and the late John H. Trench, Trin- ity University; and Alex Stanoyevitch, the University of Hawaii; and instruc- tors and students from many institutions in the United States and other coun- tries, who shared with us their experiences with the book and offered helpful suggestions.

**Other books:**

*PAINLESS ALGEBRA PDF*

The numerous suggestions, comments, and criticisms of these people greatly improved the manuscript. To all of them goes a sincere expression of gratitude. He found a number of errors in the manuscript and cheerfully performed miracles under a very tight schedule.

It was a pleasure working with him. We should also like to thank Nina Edelman, Temple University, along with Lilian Brady, for critically reading the page proofs.

Thanks also to Blaise deSesa for his help in editing and checking the solutions to the exercises. Finally, a sincere expression of thanks to Jeanne Audino, Production Ed- itor, who patiently and expertly guided this book from launch to publication; to George Lobell, Executive Editor; and to the entire staff of Prentice Hall for their enthusiasm, interest, and unfailing cooperation during the conception, design, production, and marketing phases of this edition.

Bernard Kolman bkolman mcs. Hill hill math. First, it may be your initial introduction to abstraction. Second, it is a mathematics course that may well have the greatest impact on your vocation. Unlike other mathematics courses, this course will not give you a toolkit of isolated computational techniques for solving certain types of problems.

Instead, we will develop a core of material called linear algebra by introduc- ing certain definitions and creating procedures for determining properties and proving theorems. Proving a theorem is a skill that takes time to master, so at first we will only expect you to read and understand the proof of a theorem. As you progress in the course, you will be able to tackle some simple proofs. We introduce you to abstraction slowly, keep it to a minimum, and amply il- lustrate each abstract idea with concrete numerical examples and applications.

Linear algebra is used in the everyday world to solve problems in other areas of mathematics, physics, biology, chemistry, engineering, statistics, eco- nomics, finance, psychology, and sociology. Applications that use linear alge- bra include the transmission of information, the development of special effects in film and video, recording of sound, Web search engines on the Internet, and economic analyses. Thus, you can see how profoundly linear algebra affects you.

A selected number of applications are included in this book, and if there is enough time, some of these may be covered in this course. Additionally, many of the applications can be used as self-study projects. There are three different types of exercises in this book. First, there are computational exercises. These exercises and the numbers in them have been carefully chosen so that almost all of them can readily be done by hand. A taste of this type of software is provided by the third type of exercises.

These are exercises designed to be solved by using a computer and M ATLAB, a powerful matrix-based application that is widely used in industry. The second type of exercises are theoretical.

Some of these may ask you to prove a result or discuss an idea. Mathematics uses words, not just symbols. You might have to read a particular section more than once. If you wait until the problems are explained in class, you will miss learning how to solve a problem by yourself.

You might find it helpful to work with other students on the mate- rial covered in class and on some homework problems. Each abstract idea in this course is based on previously developed ideas— much like laying a foundation and then building a house.

If any of the ideas are fuzzy to you or missing, your knowledge of the course will not be sturdy enough for you to grasp succeeding ideas.

At the end of each section we have a list of key terms; at the end of each chapter we have a list of key ideas for review, supplementary exercises, and a chapter test.

Finally, there is a glossary for linear algebra at the end of the book. Answers to the odd- numbered exercises appear at the end of the book. The Student Solutions Manual provides detailed solutions to all odd-numbered exercises, both numerical and theoretical. It can be purchased from the publisher ISBN We assure you that your efforts to learn linear algebra well will be amply rewarded in other courses and in your professional career. We wish you much success in your study of linear algebra.

The word linear is used here because the graph of the equation above is a straight line. In many applications we are given b and the constants a1 , a2 ,. A solution to a linear equation 1 is a sequence of n numbers s1 , s2 ,.

More generally, a system of m linear equations in n unknowns x1 , x2 ,. The first subscript i indi- cates that we are dealing with the ith equation, while the second subscript j is associated with the jth variable x j. In 2 the ai j are known constants.

Given values of b1 , b2 ,. A solution to a linear system 2 is a sequence of n numbers s1 , s2 ,. To find solutions to a linear system, we shall use a technique called the method of elimination.

That is, we eliminate some of the unknowns by adding a multiple of one equation to another equation. Most readers have had some experience with this technique in high school algebra courses. If we deal with two, three, or four unknowns, we shall often write them as x, y, z, and w.

In this section we use the method of elimination as it was studied in high school.

In Section 1. The rules of the trust state that both a certificate of deposit CD and a long-term bond must be used. The director determines the amount x to invest in the CD and the amount y to invest in the bond as follows: We have eliminated the unknown x. Again, we decide to eliminate x. This means that the linear system 4 has no solution. We might have come to the same conclusion from observing that in 4 the left side of the second equation is twice the left side of the first equation, but the right side of the second equation is not twice the right side of the first equation.

The importance of the procedure lies in the fact that the linear systems 5 and 8 have exactly the same solutions. System 8 has the advantage that it can be solved quite easily, giving the foregoing values for x, y, and z. This means that the linear system 9 has infinitely many solutions.

Every time we assign a value to r , we obtain another solution to 9. Since 14 and 13 have the same solutions, we con- clude that 13 has no solutions. These examples suggest that a linear system may have one solution a unique solution , no solution, or infinitely many solutions. We have seen that the method of elimination consists of repeatedly per- forming the following operations: Interchange two equations.

Multiply an equation by a nonzero constant. Add a multiple of one equation to another. It is not difficult to show Exercises T. The new linear system can then be solved quite readily. Thus we have not indicated any rules for selecting the unknowns to be eliminated. Before providing a systematic description of the method of elimination, we introduce, in the next section, the notion of a matrix, which will greatly simplify our notation and will enable us to develop tools to solve many important problems.

Consider now a linear system of two equations in the unknowns x and y: The graph of each of these equations is a straight line, which we denote by l1 and l2 , respectively.

See Figure 1. Thus we are led geometrically to the same three possibilities mentioned previously. The system has a unique solution; that is, the lines l1 and l2 intersect at exactly one point. The system has no solution; that is, the lines l1 and l2 do not intersect.

The system has infinitely many solutions; that is, the lines l1 and l2 coin- cide. Figure 1. The graph of each of these equations is a plane, denoted by P1 , P2 , and P3 , respectively. As in the case of a linear system of two equations in two un- knowns, the linear system in 16 can have a unique solution, no solution, or infinitely many solutions.

These situations are illustrated in Figure 1. For a more concrete illustration of some of the possible cases, the walls planes of a room intersect in a unique point, a corner of the room, so the linear system has a unique solution. Next, think of the planes as pages of a book. Three pages of a book when held open intersect in a straight line, the spine.

Thus, the linear system has infinitely many solutions. On the other hand, when the book is closed, three pages of a book appear to be parallel and do not intersect, so the linear system has no solution.

A, B, and C. Each product must go through two processing machines: X and Y. The products require the following times in machines X and Y: One ton of A requires 2 hours in machine X and 2 hours in machine Y.

One ton of B requires 3 hours in machine X and 2 hours in machine Y. One ton of C requires 4 hours in machine X and 3 hours in machine Y. Machine X is available 80 hours per week and machine Y is available 60 hours per week. Since management does not want to keep the expensive machines X and Y idle, it would like to know how many tons of each product to make so that the machines are fully utilized.

It is assumed that the manufacturer can sell as much of the products as is made. To solve this problem, we let x1 , x2 , and x3 denote the number of tons of products A, B, and C, respectively, to be made. This linear system has infinitely many solutions. The reader should observe that one solution is just as good as the other. There is no best solution unless additional information or restrictions are given.

## Course 311 - Abstract Algebra

Key Terms Linear equation Solution to a linear system No solution Unknowns Method of elimination Infinitely many solutions Solution to a linear equation Unique solution Manipulations on a linear system Linear system 1.

Without using the method of elimination, solve the 9. Without using the method of elimination, solve the linear system If there is, find it. A plastics manufacturer makes two types of plastic: If there is, find regular and special. Each ton of regular plastic requires it. Describe the number of points that simultaneously lie in each of the three planes shown in each part of Figure 1. A dietician is preparing a meal consisting of foods A, B, and C.

Each ounce of food A contains 2 units of protein, Describe the number of points that simultaneously lie in 3 units of fat, and 4 units of carbohydrate. Each ounce of each of the three planes shown in each part of Figure 1. Each ounce of food C contains 3 P1 P1 units of protein, 3 units of fat, and 2 units of P3 carbohydrate. If the meal must provide exactly 25 units P2 of protein, 24 units of fat, and 21 units of carbohydrate, P2 P3 how many ounces of each type of food should be used? A manufacturer makes 2-minute, 6-minute, and a b 9-minute film developers.

Each ton of 2-minute developer requires 6 minutes in plant A and 24 minutes in plant B.

Each ton of 6-minute developer requires 12 minutes in plant A and 12 minutes in plant B. Each ton P1 P2 of 9-minute developer requires 12 minutes in plant A and 12 minutes in plant B.

If plant A is available 10 hours per day and plant B is available 16 hours per day, how many tons of each type of developer can be P3 produced so that the plants are fully utilized? An oil refinery produces low-sulfur and high-sulfur fuel. If the blending trusts, with the second trust receiving twice as much as plant is available for 3 hours and the refining plant is the first trust. How utilized? Theoretical Exercises T. Show that the linear system obtained by interchanging equation in 2 by itself plus a multiple of another two equations in 2 has exactly the same solutions as equation in 2 has exactly the same solutions as 2.

Does the linear system T. Show that the linear system obtained by replacing an always have a solution for any values of a, b, c, and d? Only the numbers in front of the unknowns x1 , x2 ,. Thus we might think of looking for a way of writing a linear system without having to carry along the unknowns.

In this section we define an ob- ject, a matrix, that enables us to do this—that is, to write linear systems in a compact form that makes it easier to automate the elimination method on a computer in order to obtain a fast and efficient procedure for finding solutions.

The use of a matrix is not, however, merely that of a convenient notation. We now develop operations on matrices plural of matrix and will work with ma- trices according to the rules they obey; this will enable us to solve systems of linear equations and solve other computational problems in a fast and efficient manner.

Of course, as any good definition should do, the notion of a matrix provides not only a new way of looking at old problems, but also gives rise to a great many new questions, some of which we study in this book.

For the sake of simplicity, we restrict our attention in this book, except for Appendix A, to matrices all of whose entries are real numbers. However, matrices with complex entries are studied and are important in applications. For convenience, we focus much of our attention in the illustrative ex- amples and exercises in Chapters 1—7 on matrices and expressions containing only real numbers.

Complex numbers will make a brief appearance in Chap- ters 8 and 9. An introduction to complex numbers, their properties, and exam- ples and exercises showing how complex numbers are used in linear algebra may be found in Appendix A. When n is understood, we refer to n-vectors merely as vectors. In Chapter 4 we discuss vectors at length. The set of all n-vectors with real entries is denoted by R n.

Similarly, the set of all n-vectors with complex entries is denoted by C n. As we have already pointed out, in the first seven chapters of this book we will work almost entirely with vectors in R n. For example, plant 2 makes units of product 3 in one week. Most of the books were reedited several times with significant changes between editions , and the books were released in several parts containing different chapters e.

Book II, Algebra , was released in five parts, the first in with chapters 1, 2, and 3, and the last in containing chapter After that several new chapters to existing books as well as revised editions of existing chapters appeared until the publication of chapters 8 and 9 of Commutative Algebra in A long break in publishing activity followed, leading many to suspect the end of the publishing project.

More importantly, the first four chapters of a completely new book on algebraic topology were published in The new material from and address some references to forthcoming books in the book on Lie Groups and Algebras; there remain other such references some very precise to expected additional chapters of the book spectral theory.

The impact of Bourbaki's work initially was great on many active research mathematicians worldwide. For example:. Our time is witnessing the creation of a monumental work: Moreover this exposition is done in such a way that the common bond between the various branches of mathematics become clearly visible, that the framework which supports the whole structure is not apt to become obsolete in a very short time, and that it can easily absorb new ideas.

It provoked some hostility, too, mostly on the side of classical analysts ; they approved of rigour but not of high abstraction. Around , also, some parts of geometry were still not fully axiomatic — in less prominent developments, one way or another, these were brought into line with the new foundational standards, or quietly dropped.

This led to a gulf with the way theoretical physics was practiced. Bourbaki's direct influence has decreased over time. On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed. The Bourbaki seminar series founded in post-WWII Paris continues; it has been going on since , and contains more than items. It is an important source of survey articles , with sketches or sometimes improvements of proofs.

## MMA501 Abstract Algebra

The topics range through all branches of mathematics, including sometimes theoretical physics. The idea is that the presentation should be on the level of specialists, but should be tailored to an audience which is not specialized in the particular field. It is fairly clear that the Bourbaki point of view, while encyclopedic , was never intended as neutral.

Quite the opposite: But always through a transforming process of reception and selection—their ability to sustain this collective, critical approach has been described as "something unusual".

The following is a list of some of the criticisms commonly made of the Bourbaki approach. Pierre Cartier , a Bourbaki member between and , said that: There is also nothing about combinatorics, nothing about algebraic topology, nothing about concrete geometry. And Bourbaki never seriously considered logic.

Anything connected with mathematical physics is totally absent from Bourbaki's text. In addition, algorithms are considered off-topic and almost completely omitted.

Taking the case of locally compact measure spaces as fundamental focuses the presentation on Radon measures and leads to an approach to measurable functions that is cumbersome, especially from the viewpoint of probability theory.

Logic is treated minimally. Furthermore, Bourbaki makes only limited use of pictures in their presentation. Pierre Cartier is quoted as later saying: While several of Bourbaki's books have become standard references in their fields, some have felt that the austere presentation makes them unsuitable as textbooks. In the longer term, the manifesto of Bourbaki has had a definite and deep influence. In secondary education the new math movement corresponded to teachers influenced by Bourbaki.

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