Introductory Modern Algebra E-Book

Saul Stahl

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 726

Veröffentlichungsjahr: 2013

Beschreibung

Praise for the First Edition

"Stahl offers the solvability of equations from the historical point of view...one of the best books available to support a one-semester introduction to abstract algebra."
—CHOICE

Introductory Modern Algebra: A Historical Approach, Second Edition presents the evolution of algebra and provides readers with the opportunity to view modern algebra as a consistent movement from concrete problems to abstract principles. With a few pertinent excerpts from the writings of some of the greatest mathematicians, the Second Edition uniquely facilitates the understanding of pivotal algebraic ideas.

The author provides a clear, precise, and accessible introduction to modern algebra and also helps to develop a more immediate and well-grounded understanding of how equations lead to permutation groups and what those groups can inform us about such diverse items as multivariate functions and the 15-puzzle. Featuring new sections on topics such as group homomorphisms, the RSA algorithm, complex conjugation, the factorization of real polynomials, and the fundamental theorem of algebra, the Second Edition also includes:

• An in-depth explanation of the principles and practices of modern algebra in terms of the historical development from the Renaissance solution of the cubic equation to Dedekind's ideals
• Historical discussions integrated with the development of modern and abstract algebra in addition to many new explicit statements of theorems, definitions, and terminology
• A new appendix on logic and proofs, sets, functions, and equivalence relations
• Over 1,000 new examples and multi-level exercises at the end of each section and chapter as well as updated chapter summaries

Introductory Modern Algebra: A Historical Approach, Second Edition is an excellent textbook for upper-undergraduate courses in modern and abstract algebra.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Preface

Chapter 1: The Early History

1.1 The Breakthrough

Chapter 2: Complex Numbers

2.1 Rational Functions of Complex Numbers

2.2 Complex Roots

2.3 Solvability by Radicals I

2.4 Ruler-and-Compass Constructibility of Regular Polygons

2.5 Orders of Roots of Unity

2.6 The Existence of Complex Numbers

Chapter 3: Solutions of Equations

3.1 The Cubic Formula

3.2 Solvability by Radicals II

3.3 Other Types of Solutions

Chapter 4: Modular Arithmetic

4.1 Modular Addition, Subtraction, and Multiplication

4.2 The Euclidean Algorithm and Modular Inverses

4.3 Radicals in Modular Arithmetic

4.4 The Fundamental Theorem of Arithmetic

Chapter 5: The Binomial Theorem and Modular Powers

5.1 The Binomial Theorem

5.2 Fermat’s Theorem and Modular Exponents

5.3 The Multinomial Theorem

5.4 The Euler φ-Function

Chapter 6: Polynomials Over A Field

6.1 Fields and Their Polynomials

6.2 The Factorization of Polynomials

6.3 The Euclidean Algorithm for Polynomials

6.4 Elementary Symmetric Polynomials

6.5 Lagrange’s Solution of the Quartic Equation

Chapter 7: Galois Fields

7.1 Galois’s Construction of His Fields

7.2 The Galois Polynomial

7.3 The Primitive Element Theorem

7.4 On the Variety of Galois Fields

Chapter 8: Permutations

8.1 Permuting the Variables of a Function I

8.2 Permutations

8.3 Permuting the Variables of a Function II

8.4 The Parity of a Permutation

Chapter 9: Groups

9.1 Permutation Groups

9.2 Abstract Groups

9.3 Isomorphisms of Groups and Orders of Elements

9.4 Subgroups and Their Orders

9.5 Cyclic Groups and Subgroups

9.6 Cayley’s Theorem

Chapter 10: Quotient Groups and Their Uses

10.1 Quotient Groups

10.2 Group Homomorphisms

10.3 The Rigorous Construction of Fields

10.4 Galois Groups and the Resolvability of Equations

Chapter 11: Topics in Elementary Group Theory

11.1 The Direct Product of Groups

11.2 More Classifications

Chapter 12: Number Theory

12.1 Pythagorean Triples

12.2 Sums of Two Squares

12.3 Quadratic Reciprocity

12.4 The Gaussian Integers

12.5 Eulerian Integers and Others

12.6 What Is the Essence of Primality?

Chapter 13: The Arithmetic of Ideals

13.1 Preliminaries

13.2 Integers of a Quadratic Field

13.3 Ideals

13.4 Cancelation of Ideals

13.5 Norms of Ideals

13.6 Prime Ideals and Unique Factorization

13.7 Constructing Prime Ideals

Chapter 14: Abstract Rings

14.1 Rings

14.2 Ideals

14.3 Domains

14.4 Quotients of Rings

A. Excerpts from Al-Khwarizmi’s Solution of the Quadratic Equation1

B. Excerpts from Cardano’s Ars Magna1

C. Excerpts from Abel’s A Demonstration of the Impossibility of the Algebraic Resolution of General Equations Whose Degree Exceeds Four1

D. Excerpts from Galois’s On the Theory of Numbers1

E. Excerpts from Cayley’s The Theory of Groups1

F. Mathematical Induction

G. Logic, Predicates, Sets, and Functions

G.1 Truth Tables

G.2 Modeling Implication

G.3 Predicates and Their Negation

G.4 Two Applications

G.5 Sets

G.6 Functions

Biographies

Bibliography

Solutions to Selected Exercises

Index

Notation

Introductory Modern Algebra

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Stahl, Saul.Introductory modern algebra : a historical approach / Saul Stahl, Department of Mathematics,University of Kansas. — Second edition.   p. cm.  Includes bibliographical references and index.  ISBN 978-0-470-87616-9 (cloth)  1. Algebra, Abstract. I. Title.  QA162.S73 2013  512’.02—dc23  2013018928

Preface

IT IS COMMON KNOWLEDGE amongst mathematicians that much of modern algebra has its roots in the issue of solvability of equations by radicals. The purpose of this text is to provide the undergraduate mathematics majors and the prospective high school mathematics teachers with a one-semester introduction to modern algebra that keeps this relationship in view at all times.

Most modern algebra texts employ an axiomatic strategy that begins with abstract groups and ends with fields, ignoring the issue of solvability of equations by radicals. By contrast, we follow the paper trail from the Renaissance solution of the cubic equation to Galois’s description of his ideas. In the process, all the important concepts are encountered, each in a well-motivated manner.

One year of calculus provides all the information required for the comprehension of all the topics in this text, which has many distinguishing features:

Historical development. Students would prefer to know the real reasons that underlie the creation of the mathematical structures they encounter. They also enjoy being placed in direct contact with the works of the prime movers of mathematics. This text tries to bring them as close to the source as possible.

Finite groups and fields are rooted in some specific investigations of Lagrange, Gauss, Cauchy, Abel, and Galois regarding the solvability of equations by radicals, This text makes these connections explicit. Gauss’s proof of the constructibility of the regular 17-sided polygon is incorporated into the development, and the argument given is merely a paraphrase of that which appears in the Disquisitiones. Similarly, the proof of Theorem 8.10 is just a reorganization of that given by Abel in his paper on the quintic equation. The construction of Galois fields is accomplished in the form of a commentary on the opening pages of Galois’s paper On the Theory of Numbers which are quoted verbatim in the text. Several important documents are also included as appendices. A considerable amount of historical discussion is integrated into the development of the subject matter.

Cohesive organization. The historical development of the material allows for very little flexibility. Each chapter elucidates some of the preceding material and motivates ideas that come later. The advantage of this approach is the same as that of good motivation in general: it aids comprehension by providing the students with a framework in which to fit the various concepts they encounter. A one semester course can be constructed on the basis of Sections 1.1, 2.1–5, 3.1–2. 4.1–2, 5.1–2, 6.1–3, 7.1–3, 8.1–4, 9.1–5, & 10.1.

Figure 0.1 illustrates the author’s perception of the evolution of abstract group theory (ignoring all the geometric and much of the number-theoretic contributions). The number in the right of each box denotes the chapter in which this topic is discussed. Solid arrows correspond to connections that are treated in some depth whereas those that are displayed by dashed arrows are touched on only informally.

Figure 0.1 The genesis of the theory of finite groups.

Chapters 1 to 3 are dedicated to the formalization of the notion of solvability by radicals. Gauss’s proof of the constructibility of the regular 17-sided polygon is the capstone theorem of this part of the course. Field theory is developed in Chapters 4 to 7. The Primitive Element Theorem of Section 7.3 serves as a watershed: it unifies many of the important concepts that precede it and motivates the notion of cyclicity that comes later. Group theory is developed in Chapters 8 to 10. This begins with an explanation of the relevance of permutations to solvability by radicals, goes on to the discussion of permutation groups and abstract groups, and concludes with the description of quotient groups. Chapter 11 is meant to acquaint the students with some of the standard tools of elementary group theory.

Exercises. Each section is followed by its own set of exercises. These range from the routine to the challenging. Each chapter has an additional set of easy review exercises added to remind the students of the chapter’s main points. There are over 1,000 of these end-of-section and chapter review exercises. The answers to selected odd exercises appear at the end of the book. Most chapters are also accompanied by a collection of supplementary computer and/or mathematical projects. Some of the latter involve open questions.

Additional pedagogy. Each chapter begins with an introduction and concludes with a summary. The purposes of both the introduction and the summary are to provide the student with an overview of the chapter, and sometimes to comment on its relationship to the previous chapters. The examples are integrated into the exposition and they are highlighted by a notation in the margin. Each chapter’s new terms are listed, together with the pages on which they are defined, following that chapter’s summary.

Instructor’s manual. An instructor’s manual is available. It contains the answers to all the end of section and chapter review exercises. Some suggested homework assignments and tests are also included.

Acknowledgments

First and foremost I wish to acknowledge the substantial contributions made by Fred Galvin who rooted out several inaccuracies in the original development, improved and/or corrected many of the proofs, both in the text and the manual, suggested new exercises, and used the manuscript in his class. Thanks are also due to Todd Eisworth, Andy Magid and Phil Montgomery who also class tested the manuscript and made valuable suggestions as well as to my colleague Paul J. McCarthy who was kind enough to lend me both an ear and his algebraic expertise. It remains to gratefully acknowledge the efforts of Jessica Downey, Steve Quigley, Rosalyn Farkas, and Lisa Van Horn of John Wiley & Sons on behalf of this book.

June 1996

Preface to the Second Edition

Surprisingly, it turned out that the historical approach could be used to teach ring theory as well. The point of departure is the Theorem of Pythagoras, viewed as a diophantine equation. Chapter 12 begins there and goes on to Fermat’s characterization of primes that are the sum of two square integers. From there we go on to quadratic reciprocity and the Gaussian integers. The question of Gaussian primes is natural and some attention is given to variant number systems with radicals or . The chapter ends with a discussion of Kummer’s decision to redefine the notion of primality.

Quadratic fields, quadratic integers, and ideals are defined and the arithmetic of ideals is explored in Chapter 13. It is shown that the arithmetic of ideals does possess the unique factorization property. Finally, Chapter 14 discusses rings and ideals in the abstract manner of today.

The author’s understanding of the low level algebraic number theory in Chapter 13 comes from reading one of Keith Conrad’s many expository monographs. The solutions to the selected exercises in Chapters 13 and 14 were derived by Grant Serio and are included with his permission. Katie Ballentine, Annika Denkert, and Mark Hunacek debugged portions of the manuscript, which was expertly typeset by Lon Mitchell.

June 2013

Saul StahlLawrence, Kansas

Chapter 1

THE EARLY HISTORY

THIS CHAPTER CONTAINS an informal account of the early history of the issue of solvability of equations of degrees one, two, and three in a single unknown. The formulas that provide the solutions lead in a natural way to the discussion of the origins of complex numbers. We also take this opportunity to review some well-known information about the quadratic equation.

1.1 The Breakthrough

There is a general agreement among historians of mathematics that modern mathematics came into being in the mid sixteenth century when the combined efforts of the Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano produced a formula for the solution of cubic equations. For the first time ever west European mathematicians succeeded in cracking a problem whose solution eluded the best mathematical minds of antiquity. Archimedes, one of the greatest mathematicians, scientists, and engineers of all times, had solved some cubic equations in terms of the intersections of a suitable parabola and hyperbola. Omar Khayyam, one of the most prominent of the Arab mathematicians and poets, also expended much effort on his geometrical solutions of special cases of the cubic equation but could not find the general formula. However, the significance of this accomplishment of the Renaissance mathematicians is not limited to the difficulty of the problem that was solved. We shall try to show how the issues raised by this solution eventually led to the creation of modern algebra and the discovery of mathematical landscapes that were undreamt of, even by such imaginative investigators as Archimedes and Khayyam.

The interest in algebraic equations goes back to the beginnings of written history. The Rhind Mathematical Papyrus, found in Egypt circa 1856 is a copy of a list of mathematical problems compiled some time during the second half of the nineteenth century BCE, or possibly even earlier. The twenty-fourth of these problems reads: “A quantity and its 1/7 added become 19. What is the quantity?” In other words, what is the solution to the equation

The method employed by the scribe has come to be known as the method of false position. He replaces the unknown by 7 and observes that

From this he concludes that the correct answer is obtained upon multiplying the first guess of 7 by 19/8:

Interestingly enough, the scribe does double check his solution by substituting it into the original problem and verifying that

We will not discuss the merits and limitations of the method of false position except to note that the idea of obtaining a correct solution to an equation by starting out with a possibly false guess and then modifying that guess has been refined into powerful techniques for finding numerical solutions, one of which will be described in Section 3.3. We do, however, wish to point out that the general first-degree equation is today defined as

and that the rules of algebra yield

as its unique solution.

The Mesopotamian mathematicians of that time could solve much more intricate equations, and had in fact already developed techniques for solving what we nowadays call quadratic equations. These techniques employed the geometrical method of “completing the square.” The Greeks, Indians, and Arabs all were aware of this method, having either derived them independently or perhaps learnt them from their predecessors and/or neighbors. In the ninth century the Persian mathematician al-Khwarizmi wrote the book Hisab al-jabr w’al-muqa-balah in which he carefully explained a compendium of algebraic techniques learnt from several past civilizations. The clarity of his exposition won both him and his book immortality in that the portion al-jabr of the title evolved into the word algebra, and the author’s name is the source of the word algorithm. An excerpt from this book expounding the solution to the quadratic equation

appears in Appendix A. The modern solution of the quadratic also relies on the completion of the square. The general quadratic equation has the form

(1.1)

and its solutions are found by first factoring out the coefficient a and then completing the rest to a perfect square. Thus, we first divide Equation 1.1 through by a to obtain the equation

(1.2)

The left side of Equation 1.2 is then transformed to a near perfect square:

The original quadratic equation has thus been transformed to

or

Hence the general quadratic equation, Equation 1.1, has the two solutions

from which it follows that it is easy to construct a quadratic equation whose roots are prespecified. As we will have several occasions to refer to these identities later, they are stated as a proposition whose proof is relegated to Exercise 1.1.14.

Proposition 1.3 For any two numbers r and s the quadratic equation

has r and s as its roots.

It is reasonable at this point to raise the ante and ask for a formula that will yield the solution of the general cubic equation

(1.4)

There are indications that the Mesopotamians already tried to systematize the search for solutions of cubic equations, and we know for a fact that the Greeks attempted the same. As was mentioned above, the final breakthrough did not occur until the middle of the sixteenth century when it was shown that a solution of the equation

is given by the expression

(1.5)

As we shall see later, very little additional work is required to pass from this formula on to a formula for the general cubic equation (Equation 1.4), and so Formula 1.5 can be considered as the crucial step, even though it does not yield the solution to the most general cubic equation.

In analogy with the ancient solutions of the quadratic, this solution was obtained by a geometrical process of completing the cube. Excerpts from Cardano’s description of the solution are contained in Appendix B. A modern derivation of this formula appears in Chapter 3, and we restrict ourselves here to the examination of some instructive applications of Formula 1.5. Surprisingly, this formula raises some very interesting questions.

(1.6)

two more solutions of the original equation are obtained by solving the quadratic

As the solutions of this quadratic are −2 ± , we are faced with the question of which of the three numbers 4 or −2 ± is disguised as Expression 1.6. Moreover, this complicated expression involves square roots of negative numbers, in other words, imaginary quantities, whereas 4 and −2 ± are all real numbers. This apparent paradox was resolved by Bombelli who simplified Expression 1.6 by setting

and similarly

Consequently,

The solution to the cubic equation is the context within which imaginary numbers were first discussed by mathematicians. Cardano toyed with them and then rejected them as useless. Bombelli gave them more credence, but it wasn’t until about 200 years later that the work of Leonhard Euler, Pierre-Simon de Laplace, and later that of Carl Friedrich Gauss, Augustin-Louis Cauchy, and Niels Abel turned the complex number system, consisting of both the real and imaginary numbers, into an indispensable tool for mathematical researchers.

Exercises 1.1

Chapter Summary

This introductory chapter was used to briefly review the solutions of the first- and second-degree equations in a single unknown. The history of the solution of the cubic equation was also discussed and the relationship of this formula to the complex number system was examined.

Chapter Review Exercises

Mark the following true or false.

1. Every real number is the solution of some equation.

2. Every pair of real numbers is the solution set of some quadratic equation.

3. Every equation has at least one solution.

New Terms

cubic equation, 4

method of false position, 2

first-degree equation, 2

quadratic equation, 3

Chapter 2

COMPLEX NUMBERS

THROUGHOUT HISTORY, the introduction of new numbers has been greeted with considerable resistance on the part of mathematicians. Legend has it that the discoverer of irrational numbers was rewarded by being drowned by his fellow Greeks. Be that as it may, the fact is that these numbers have been tagged with the pejorative label of irrational, a word which, when used in nonmathematical contexts, has definite derogatory connotations. The same, of course, applies to the negative numbers. The imaginary numbers have been cursed with what is arguably the worst nomenclature in mathematics. Given the considerable difficulties that the average students face in learning the rigorous discipline of mathematics, can they be blamed for balking at having to contend with quantities that mathematicians themselves admit are imaginary?

The best way to overcome people’s resistance to a new concept is to convince them of its utility. Accordingly, it will be shown that the widening of our field of operations to include the complex numbers greatly enhances the power of the Ferro-Tartaglia-Cardano cubic formula. Next, the complex numbers will be used to solve some ruler-and-compass construction problems of plane geometry. Only in this chapter’s last section will the issue of the existence of the complex numbers be addressed.

2.1 Rational Functions of Complex Numbers

The multiplication of complex numbers also resembles that of polynomials, except that each occurrence of i2 is replaced by −1. Thus,

The division of complex numbers mimics the well-known process of rationalizing denominators. Thus,

Figure 2.1 Complex addition

and

Consequently, PR || OQ and QR || OP and so OPRQ is a parallelogram. Thus we see that the addition of complex numbers resembles that of vectors. These considerations are summarized as follows.

Proposition 2.1 Let O denote the origin of the Cartesian plane and let P and Q be the Cartesian representations of the complex numbers a + bi and c + di, respectively. If the sum of the two complex numbers is represented by the point R, then the quadrilateral OPRQ is a parallelogram.

Figure 2.2 The argument and the modulus

Figure 2.3 Some complex numbers

whereas

The trigonometric formulas for the functions of the sums of two angles yield

and .

Angles whose measures differ by integer multiples of 2π are considered to be identical. Thus, the number i has all the following as its arguments:

This is necessitated by such observations as the fact that

This observation can be put to good use in computing large powers of complex numbers. Consider the problem of computing (1 + i)100. By Corollary 2.3,

Proof. See Exercise 2.1.27.

Exercises 2.1

Find the argument and modulus of each of the complex numbers in Exercises 2.1.1 to 2.1.4

1. 2 + 3i

2. 3 − 2i

3. −3 − 4i

4. −1 + 7i

Express the complex quantities in Exercises 2.1.5 to 2.1.21 in the form a + bi, where a and b are real numbers.

5. (2 + 3i) + (5 − i)

6. (17 − 3i) + (2 + 3i)

7. (2 + 3i)(5 − i)

8. (17 − 3i)−(2 + 3i)

9. (2 + 3i)(5 − i)

10. (17 − 3i)(2 + 3i)

11. (2 + 3i)/(5 − i)

12. (17 − 3i)/(2 + 3i)

13.

14. (a + bi)/(a − bi) − (a − bi)/(a + bi)

15. (2 − i)2/(1 + i)

16. (1 + i)4

17. (1 − 2i)4

18. (1 − i)63

19. i4,321

20. ((1 − i)/(1 + i))127

21. i4n+3 (n is an integer)

Solve the equations in Exercises 2.1.22 to 2.1.25 for z and w:

2.2 Complex Roots

Another square root of i is of course

An alternate method for arriving at −j is to recall that arg(i) could also have been taken as 5π/2, in which case we obtain the square root

It can be easily verified by direct calculations that

This procedure yields three different values for, . For, taking as the argument of 1 the successive values of 0, 2π, 4π, 6π,…, each of the elements of must have as its argument one of the values 0, 2π/3, 4π/3, 2π,…. The modulus of 1 being 1, it follows that the modulus of must be the real cube root of 1 which is also 1. Hence we get as our cube roots of 1 the following numbers:

It is clear that this list of cube roots of 1 will cycle through the same three values. The second root, the one with argument 2π/3, is denoted by ω. By Corollary 2.3, the third root, having double the argument of ω and the same modulus of 1, equals ω2. Note that the Cartesian representations of these three complex cube roots of 1 form an equilateral triangle in the Cartesian plane (Figure 2.4).

Figure 2.4 Complex roots of unity

The same procedure yields all the n-th roots of unity (roots of 1) for each positive integer n.

The n-th root of unity

and is independent of k, it follows that the elements of form a regular n-gon that is centered at the origin. This fact is crucial for the next section, and so we state it as a proposition.

Proposition 2.6 For any fixed integer n ≥ 3, the Cartesian representations of the elements of form the vertices of a regular n-gon.

it follows that

the elements of which form the vertices of a square (Figure 2.4).

it follows that

the elements of which form the vertices of the regular hexagon of Figure 2.4.

The following proposition is both a natural extension and a corollary of Theorem 2.5. Since our main interest lies in the roots of unity, its proof is omitted and relegated to Exercise 2.2.18, The subsequent example clarifies the purport of the proposition.

Proposition 2.7 Let n be a positive integer and z any nonzero complex number with argument θ. If

then

where denotes the common modulus of all the elements of .

Since −1 + i has modulus 2 and argument 3π/4, and since

it follows that

It should be noted here that calculators are very handy in computing complex roots too. Thus, to compute we note that

and

Hence, if we set

then

We conclude this section with a curious fact that will shortly prove unexpectedly useful.

Proposition 2.8 For any fixed integer n > 1, the sum of the elements of is 0.

Proof. Let ζ be the first n-th root of unity. By Theorem 2.5, the elements of can be listed as 1, ζ, ζ2,…, ζn−1. The formula for geometric progressions now yields

Exercises 2.2

Express each of the elements of the sets in Exercises 2.2.1 to 2.2.12 in the form a + bi, where a and b are real numbers.