Abstract Algebra

by W. E. Deskins, Mathematics

Overview

This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena.
In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diophantine problems, and congruences. Chapters 6 through 9 examine groups, rings, domains, fields, polynomial rings, and quadratic domains.
Chapters 10 through 13 cover modular systems, modules and vector spaces, linear transformations and matrices, and the elementary theory of matrices. The author, Professor of Mathematics at the University of Pittsburgh, includes many examples and, at the end of each chapter, a large number of problems of varying levels of difficulty.

Product Details

ISBN-13:	9780486688886
Publisher:	Dover Publications
Publication date:	01/16/1996
Series:	Dover Books on Mathematics Series
Edition description:	Reprint
Pages:	640
Product dimensions:	5.42(w) x 8.48(h) x 1.25(d)

Read an Excerpt

ABSTRACT ALGEBRA

---

By W.E. DESKINS

Dover Publications, Inc.

Copyright © 1992 W.E. Deskins
All rights reserved.
ISBN: 978-0-486-15846-4

---

CHAPTER 1

A Common Language

Our aim in this book is basically a very simple one: to show how the familiar methods and ideas used in the analysis of the system of "whole numbers" have been and are being extended and generalized to enable more diverse mathematical systems to be analyzed. In particular the concept of factorization or decomposition into "simpler" pieces underlies essentially all of our efforts. However, before we can proceed with this program of study we must try to establish a common starting point and path of approach. Even before doing this we must agree upon some basic assumptions, terms, and notations, and these are set down in this chapter.

Since the student will have encountered most of this material previously in one form or another he should strive to maintain contact between the various formulations by constructing examples, doing problems, asking questions of his instructors and fellow students, and reading other accounts of the material. (A collection of problems is included in each section, and a suggested reading list is included at the end of each chapter.)

Apart from the assumptions and agreements which are discussed in the following sections, we shall assume that there is a general understanding of the logical reasoning involved in proceeding from hypotheses to conclusions. Although we shall at times discuss the various forms in which a theorem can be stated and will always try to set forth the important features of each proof, neither a formal treatment of logic nor the use of formal logic appears in this book.

1.1 Sets

Any discussion must involve words and ideas which either have been agreed upon as undefined and undefinable or have been defined by using such primitives. These undefined terms, which can be somewhat arbitrarily selected, take on meaning from the assumptions or axioms which describe the undefined objects by stating relations between them. The axioms naturally are used to prove conclusions or theorems concerning the undefined terms and the objects which have been defined by using the undefined terms. In this manner a theory is developed.

We begin with the undefined notions of set and of set membership. Various words such as class, family, collection, and aggregate, are used interchangeably with the word set, and the members of a set are generally referred to as the elements of the set.

Intuitively a set is a collection of objects called the elements of the set. This sentence does not define the concepts, however; they are undefined terms in the common language we assume throughout this book.

Whenever feasible a set is denoted by a capital letter and an element of a set by a small letter. Since a set can consist of a collection of other sets it is apparent that this convention cannot always be observed. The notation

a [member of] A

indicates that the element a is a member of the set A and is sometimes read as "a belongs to A." The notation

a [not member of] A

indicates that the element a is not a member of the set A.

Particular sets are usually specified in one of two ways. Either the elements of the set are listed within a set of braces, or a property is stated which the elements of the set and only the elements of the set satisfy. The only word of explanation needed for the first method is that the order in which the elements are listed is of no importance. This independence from order of listing is due to a very basic assumption.

Axiom of Extent. The set A equals the set B, written A = B, if every element of A is an element of B and every element of B is an element of A.

Thus a set is completely known if its elements are known, and the two methods of specifying a set are merely statements that the elements of a set are known if they can be listed or if a characterizing property can be found. The notation for listing the elements (i.e., braces) has already been mentioned. A slightly more elaborate notation is frequently used for the second, namely,

{x: (condition or conditions which x must satisfy)},

which is read "the set of all elements x which satisfy the condition or conditions."

Note. An element x and the set {x} consisting of the element x should not be confused with one another.

Example 1. Let A be the set A = {a, b, c, d, e}. Then the set B = {a, e} can also be described using the second method; viz.,

B = {x: x [member of] A and x is a vowel}.

Thus, "B is the set of all elements from A which are also vowels."

Now let C be the set {b, c, d}. What is meant by

D = {x: x [member of] C and x is a vowel}?

Clearly in adopting this notation we implicitly accept the axiom that for each collection of conditions there is a set whose elements are just the objects meeting those conditions. So we are led to accept the existence of a set containing no elements, the empty or void set. We use the symbol [empty set] to denote this convenient set. Then the above statement means that D = [empty set].

In Example 1 the elements of the set B are also elements of the set A, suggesting that the set B is "contained in" A. We formalize this as a definition.

Definition 1.1. A set T is a subset of a set S if and only if every element of T is also an element of S. A set T is a proper subset of a set S if and only if T is a subset of S but S is not a subset of T.

If T is a subset of S we write

T [subset or equal to] S or S [contains or equal to] T,

and if T is a proper subset of S we write

T [subset] S or S [contains] T.

This is completely analogous to the familiar notation for inequality.

Suppose we list the subsets of the set A = {x, y}. They are {x}, {y}, {x, y}, and [empty set], and if the student is surprised that A and [empty set] are subsets of A, he need only check the Definition to see that this is indeed the case. Certainly every element of A and every element of [empty set] (which contains no elements) are in A.

An important use of the concept of subset stems from the following elementary theorem. In our subsequent work we shall often want to prove that two sets are equal, and we will generally use a two-pronged attack based on this result.

Theorem 1.1. The set S equals the set T if and only if S is a subset of T and T is a subset of S.

First we remark that there are actually two theorems here, or there would be if the results were written in the conventional "If ..., then ..." form. Recasting the material in this form yields

Theorem 1.1a. If S [subset or equal to] T and R [subset or equal to] S, then S = T.

Theorem 1.1b. If S = T, then S [subset or equal to] T and T [subset or equal to] S.

The first statement is the one frequently used, and the proof is quite easy. We are given that S [subset or equal to] T and T [subset or equal to] s, statements which were defined to mean, respectively, that every element of S is an element of T and every element of T is an element of S (undefined terms). Then from the Axiom of Extent we conclude that S = T.

The second statement is proved similarly.

The notion of a subset provides us with a means of comparing sets, or ranking sets. There are also various ways of combining sets to form other sets, and by defining these in terms of our undefined concepts and studying them we build our (intuitive) theory of sets.

Definition 1.2

Union: S [union[ T = {a: a [member of] S or a [member of] T}.

Intersection: S [intersection] T = {a: a [member of] S and a [member of] T}.

Difference: S - T = {a: a [member of] S and a [not member of] T}.

In other words, the set S [union] T (read "S union T") is the collection of those elements which belong to either or both S and T; the set S [intersection] T (read "S intersection T") consists of just those elements which lie in both S and T; and the set S - T (read "S minus T") is the collection of elements of S which are not in T. We can illustrate these with the diagrams in Fig. 1.1, in which it is understood that the set S consists of the points inside the square and the set T the points inside the triangle.

Such diagrams, often called Venn diagrams, are useful visual aids to understanding relations between sets. Although they cannot be used to prove statements, the appropriate diagram can sometimes help one discern just what must be or has been proved. The diagrams sometimes suggest theorems, too. The diagrams in Fig. 1.1make plausible the following theorems:

Theorem 1.2. (S [intersection] T) [union] (S - T) = S.

Theorem 1.3. (S - T) [union] T = S [union] T.

Theorem 1.4. S [subset or equal to] S [union] T.

Theorem 1.5. (S - T) [intersection] T = [empty set].

Theorem 1.6. S - T = S - (S [intersection] T).

Theorem 1.7. S - T and T - S are in general different sets. And there are others, too.

In proving, or in attempting to prove, the validity of such statements we can use the axioms which have been assumed, the Definitions which have been made, and the theorems which have already been proved. (At the moment this collection is a small one.) As an example we shall prove Theorem 1.2.

Theorem 1.2. If S and T are sets, then

(S [intersection] T) [union] (S - T) = S.

In proving this we first use the Definitions of intersection, union, and difference to establish the two conclusions

(1.1) (S [intersection] T) [union] (S - T) [subset or equal to] S,

(1.2) (S [intersection] T) [union] (S - T) [contains or equal to] S,

Then the conclusion of Theorem 1.2 follows from Theorem 1.1. For conclusion (1.1) we need only note that the sets S [intersection] T and S - T are both subsets of S. To establish conclusion (1.2) we observe that an element s of S is either an element of T or it is not an element of T. If s [member of] T, then

s [member of] S [intersection] T;

if s [not member of] T, then

s [member of] S - T.

(Continues...)

---

Excerpted from ABSTRACT ALGEBRA by W.E. DESKINS. Copyright © 1992 W.E. Deskins. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

---

Table of Contents

PREFACE
1. A COMMON LANGUAGE
  1.1. Sets
  1.2. "Ordered pairs, products, and relations"
  1.3. Functions and mappings
  1.4. Binary operations
  1.5. Abstract systems
  1.6. Suggested reading
2. THE BASIC NUMBER SYSTEMS
  2.1 The natural number system
  2.2 Order and cancellation
  2.3. Well-ordering
  2.4. Counting and finite sets
  2.5. The integers defined
  2.6. Ordering the integers
  2.7. Isomorphic systems and extensions
  2.8. Another extension
  2.9. Order and density
  2.10.
• The real number system
  2.11. Power of the abstract approach
  2.12. Remarks
  2.13. Suggested reading
3. DECOMPOSITIONS OF INTEGERS
  3.1. Divisor theorem
  3.2. Congruence and factors
  3.3. Primes
  3.4. Greatest common factor
  3.5. Unique factorization again
  3.6. Euler's totient
  3.7. Suggested reading
4.
• DIOPHANTINE PROBLEMS
  4.1. Linear Diophantine equations
  4.2. More linear Diophantine equations
  4.3. Linear congruences
  4.4. Pythagorean triples
  4.5. Method of descent
  4.6. Sum of two squares
  4.7. Suggested reading
5. ANOTHER LOOK AT CONGRUENCES
  5.1. The system of congruence classes modulo m
  5.2. Homomorphisms
  5.3. Subsystems and quotient systems
  5.4.
• System of ideals
  5.5.
• Remarks
  5.6. Suggested reading
6. GROUPS
  6.1. Definitions and examples
  6.2. Elementary properties
  6.3. Subgroups and cyclic groups
  6.4. Cosets
  6.5. Abelian groups
  6.6.
• Finite Abelian groups
  6.7.
• Normal subgroups
  6.8.
• Sylow's theorem
  6.9.
• Additional remarks
  6.10. Suggested reading
7. "RINGS, DOMAINS, AND FIELDS"
  7.1. Definitions and examples
  7.2. Elementary properties
  7.3. Exponentiation and scalar product
  7.4. Subsystems and characteristic
  7.5. Isomorphisms and extensions
  7.6. Homomorphisms and ideals
  7.7. Ring of functions
  7.8. Suggested reading
8. POLYNOMIAL RINGS
  8.1. Polynomial rings
  8.2. Polynomial domains
  8.3. Reducibility in the domain of a field
  8.4. Reducibility over the rational field
  8.5. Ideals and extensions
  8.6. Root fields and splitting fields
  8.7.
• Automorphisms and Galois groups
  8.8.
• An application to geometry
  8.9.
• Transcendental extensions and partial fractions
  8.10. Suggested reading
9.
• QUADRATIC DOMAINS
  9.1. Quadratic fields and integers
  9.2. Factorization in quadratic domains
  9.3. Gaussian integers
  9.4. Ideals and integral bases
  9.5. The semigroup of ideals
  9.6. Factorization of ideals
  9.7. Unique factorization and primes
  9.8. Quadratic residues
  9.9. Principal ideal domains
  9.10. Remarks
  9.11. Suggested reading
10.
• MODULAR SYSTEMS
  10.1. The polynomial ring of J/(m)
  10.2. Zeros modulo a prime
  10.3. Zeros modulo a prime power
  10.4. Zeros modulo a composite
  10.5. Galois fields
  10.6. Automorphisms of a Galois field
  10.7 Suggested reading
11. MODULES AND VECTOR SPACES
  11.1. Definitions and examples
  11.2. Subspaces
  11.3. Linear independence and bases
  11.4. Dimension and isomorphism
  11.5. Row echelon form
  11.6. Uniqueness
  11.7. Systems of linear equations
  11.8. Column rank
  11.9. Suggested reading
12. LINEAR TRANSFORMATIONS AND MATRICES
  12.1. Homomorphisms and linear transformations
  12.2. Bases and matrices
  12.3. Addition
  12.4. Multiplication
  12.5. Rings of linear transformations of matrices
  12.6. Nonsingular matrices
  12.7. Change of basis
  12.8.
• Ideals and algebras
  12.9 Suggested reading
13. ELEMENTARY THEORY OF MATRICES
  13.1. Special types of matrices
  13.2. A factorization
  13.3. On the right side
  13.4. Over a polynomial domain
  13.5. Determinants
  13.6. Determinant of a product
  13.7. Characteristic polynomial
  13.8. Triangularization and diagonalization
  13.9. Nilpotent matrices and transformations
  13.10. Jordan form
  13.11. Remarks
  13.12. Suggested reading
GENERAL REFERENCES
INDEX