# ABSTRACT ALGEBRA: STUDY GUIDE

### John A. Beachy

ISBN 1-49357-411-6, © 2013, 268 pages, \$29.95 list

FEATURES

USER'S MANUAL

PREFACE

John Beachy,
Dept. of Mathematical Sciences,
Northern Illinois University,
DeKalb, Illinois 60115
TEL: 815 / 753-0567

This site dates from 1/2014.

### FEATURES

One of the important features of Abstract Algebra is the introduction of abstract concepts only after a careful study of important examples. Chapters One and Two of the Study Guide include solved problems on number theory, functions, and permutations. These provide important examples (permutation groups, cyclic groups, and other groups) for the study of groups, which begins in Chapter Three. Similarly, Chapter Four has problems on polynomials which precede the introduction of rings in Chapter Five.

The Study Guide only provides supplementary problems, and does not include answers to exercises in the textbook. But this means that students now have even more examples to work with. The solutions to the problems contain numerous comments, clearly labeled, so that it becomes apparent that they are not part of a proof.

Finally, there are additional problems to work on that may or may not have a hint or an answer. That's just to keep the studying a little bit more interesting!

### USER'S MANUAL

The Study Guide contains over 600 problems. More than 300 of these have detailed solutions, while about 100 more have either an answer or a hint. The problems are different from the exercises in the third edition of Abstract Algebra (by Beachy and Blair).

The ideal way to use the study guide is to work on a solved problem, and if you get stuck, just uncover enough of the solution to get started again. If the number of problems looks too daunting, and you already have a lot of other homework problems, I hope that you can also learn a lot by just reading some solutions. A compromise is to choose a variety of solved problems, ones that look interesting, work seriously on them, and then just read the rest of the solved problems in the section.

Because it is so tempting to peek at the solutions, they are separated from the statements of the problems, and are printed in the second half of the Study Guide. You may also find it valuable to work on problems listed under the ``More Problems'' heading that have an answer or a hint, indicated by a dagger symbol.

There is a partial index to the solved problems, which groups many of them into some general categories. Sometimes a problem is introduced in one section, and then appears again in a later section when additional results make the solution easier. The index lets you find those problems, and others that may connect back to an earlier section.

Important Note: Can this study guide be useful even if you are using a textbook that is different from Abstract Algebra? Yes, I hope so! There are occasional references to exercises from our textbook, so you might need to solve one of them to complete a solution to a problem in the Study Guide. Otherwise, the Study Guide can be used independently of Abstract Algebra, provided that you own a textbook that has proofs of the standard theorems in the subject.

More important Note: If you own a copy of the third edition of Abstract Algebra, you don't need to purchase the Study Guide, since you already have access to all of the definitions and theorems needed in the solutions of the problems. Another version of this study guide is available, gratis, on the web site given below. It contains the same problems and solutions; all that's missing is the definitions and theorems from the textbook. Web site: A Study Guide for Beginners

PREFACE | USER'S MANUAL

PART I: DEFINITIONS, THEOREMS, COMMENTS, AND PROBLEMS

Chapter 1: INTEGERS | Divisors | Primes | Congruences | Integers Modulo n | Review Problems || 18 pages

Chapter 2: FUNCTIONS | Functions | Equivalence Relations | Permutations | Review Problems || 16 pages

Chapter 3: GROUPS | Definition of a Group | Subgroups | Constructing Examples | Isomorphisms | Cyclic Groups | Permutation Groups | Homomorphisms | Cosets, Normal Subgroups, and Factor Groups | Review Problems || 42 pages

Chapter 4: POLYNOMIALS | Fields; Roots of Polynomials | Factors | Existence of Roots | Polynomials over Z, Q, R, and C | Review Problems || 20 pages

Chapter 5: COMMUTATIVE RINGS | Commutative Rings; Integral Domains | Ring Homomorphisms | Ideals and Factor Rings | Quotient Fields | Review Problems || 20 pages

Chapter 6: FIELDS | Algebraic Elements | Finite and Algebraic Extensions | Geometric Constructions || 8 pages

PART II: SOLUTIONS, ANSWERS, AND HINTS

Chapter 1: INTEGERS | Divisors | Primes | Congruences | Integers Modulo n | Review Problems || 22 pages

Chapter 2: FUNCTIONS | Functions | Equivalence Relations | Permutations | Review Problems || 14 pages

Chapter 3: GROUPS | Definition of a Group | Subgroups | Constructing Examples | Isomorphisms | Cyclic Groups | Permutation Groups | Homomorphisms | Cosets, Normal Subgroups, and Factor Groups | Review Problems || 44 pages

Chapter 4: POLYNOMIALS | Fields; Roots of Polynomials | Factors | Existence of Roots | Polynomials over Z, Q, R, and C | Review Problems || 16 pages

Chapter 5: COMMUTATIVE RINGS | Commutative Rings; Integral Domains | Ring Homomorphisms | Ideals and Factor Rings | Quotient Fields | Review Problems || 20 pages

Chapter 6: FIELDS | Algebraic Elements | Finite and Algebraic Extensions | Geometric Constructions || 4 pages

BIBLIOGRAPHY | INDEX OF SYMBOLS | INDEX OF SOLVED PROBLEMS | INDEX OF TERMS

268 pages, over 600 problems

### PREFACE

I first taught an abstract algebra course in 1968, using Herstein's Topics in Algebra. It's hard to improve on his book; the subject may have become broader, with applications to computing and other areas, but Topics contains the core of any course. Unfortunately, the subject hasn't become any easier, so students meeting abstract algebra still struggle to learn the new concepts, especially since they are probably still learning how to write their own proofs.

This "study guide" is intended to help students who are beginning to learn about abstract algebra. Instead of just expanding the material that is already written down in our textbook, Abstract Algebra, I decided to try to teach by example, by writing out solutions to problems. I've tried to choose problems that would be instructive, and in quite a few cases I've included comments to help the reader see what is really going on. Of course, the study guide isn't a substitute for a good teacher, or for the chance to work together with other students on some hard problems.

In the two semester course that I have been teaching, I have usually covered the material in Abstract Algebra through Section 6.3. This study guide is focused on those chapters. The textbook's website, given below, also contains a study guide to Chapters 7 and 8 of the text, called A Review of Groups and Galois Theory.

I hope that you will find the subject as interesting and as challenging as I have.

John A. Beachy, DeKalb, Illinois, June 2013

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