A book of abstract algebra

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Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. Cover A BOOK OF ABSTRACT ALGEBRA Copyright © 1982 by McGraw-Hill, ISBN 0-07-050130-0 QA162.P56 512'.02 Dedication CONTENTS PREFACE ACKNOWLEDGMENTS CHAPTER ONE: WHY ABSTRACT ALGEBRA? ORIGINS THE MODERN AGE The Algebra of Matrices Boolean Algebra Algebraic Structures AXIOMS AND MEN THE AXIOMATICS OF ALGEBRA ABSTRACTION REVISITED CHAPTER TWO: OPERATIONS EXERCISES A. Examples of Operations B. Properties of Operations C. Operations on a Two-Element Set CHAPTER THREE: THE DEFINITiON OF GROUPS EXERCISES A. Examples of Abelian Groups B. Groups on the Set R x R C. Groups of Subsets of a Set D. A Checkerboard Game K A Coin Game F. Groups in Binary Codes CHAPTER FOUR: ELEMENTARY PROPERTiES OF GROUPS EXERCISES A. Solving Equations in Groups B. Rules of Algebra in Groups C. Elements which Commute D. Group Elements and Their Inverses E. Counting Elements and Their Inverses F. Constructing Small Groups G. Direct Products of Groups H. Powers and Roots of Group Elements CHAPTER FIVE: SUBGROUPS EXERCISES A. Recognizing Subgroups B. Subgroups of Groups of Functions C. Subgroups of Abelian Groups D. Subgroups of an Arbitrary Group E. Generators of Groups F. Groups Determined by Generators and Defining Equations G. Cayley Diagrams CHAPTER SIX: FUNCTiONS EXERCISES A. Examples of Injective and Surjective Functions B. Functions on R and 1 C. Functions on Arbitrary Sets and Groups D. Composite Functions E. Inverses of Functions F. Functions on Finite Sets G. Some General Properties of Functions CHAPTER SEVEN: GROUPS OF PERMUTATIONS EXERCISES A. Computing Elements of S5 B. Examples of Groups of Permutations C. Groups of Permutations of R D. A Cyclic Group of Permutations E. A Subgroup of S_R F. Symmetries of Geometric Figures G. Symmetries of Polynomials H. Properties of Permutations of a Set A CHAPTER EIGHT: PERMUTATIONS OF A FINITE SET EXERCISES A. Practice in Multiplying and Factoring Permutations B. Powers of Permutations C. Even and Odd Permutations D. Disjoint Cycles E. Conjugate Cycles F. Order of Cycles G. Even/Odd Permutations in Subgroups of Sn H. Generators of An and Sn CHAPTER NINE: 1SOMORPHISM EXERCISES A. Isomorphism Is an Equivalence Relation among Groups B. Elements Which Correspond under an Isomorphism C. Isomorphism of Some Finite Groups D. Separating Groups into Isomorphism Classes F. Isomorphism of Infinite Groups F. Isomorphism of Groups Given by Generators and Defining Equations G. Isomorphic Groups on the Set R H. Some General Properties of Isomorphism I. Group Automorphisms J. Regular Representation of Groups CHAPTER TEN: ORDER OF GROUP ELEMENTS EXERCISES A. Laws of Exponents B. Examples of Orders of Elements C. Elementary Properties of Order D. Further Properties of Order E. Relationship between ord(ah), ord(a), and ord(h) F. Orders of Powers of Elements G. Relationship between ord(a) and ord(aC) H. Relationship between the Order of a and the Order of any kth Root of a CHAPTER ELEVEN: CYCLIC GROUPS EXERCISES A. Examples of Cyclic Groups B. Elementary Properties of Cyclic Groups C. Generators of Cyclic Groups D. Elementary Properties of Cyclic Subgroups of Groups E. Direct Products of Cyclic Groups F. kth Roots of Elements in a Cyclic Group CHAPTER TWELVE: PARTITIONS AND EQUIVALENCE RELATIONS EXERCISES A. Examples of Partitions B. Examples of Equivalence Relations C. Equivalence Relations and Partitions of R x R D. Equivalence Relations on Groups E. General Properties of Equivalence Relations and Partitions CHAPTER THIRTEEN: COUNTING COSETS EXERCISES A. Examples of Cosets in Finite Groups B. Examples of Cosets in Infinite groups C. Elementary Consequences of Lagrange's Theorem D. Further Elementary Consequences of Lagrange's Theorem E. Elementary Properties of Cosets F. Survey of All Six-Element Groups G. Survey of All 10-Element Groups H. Survey of All Eight-Element Groups I. Conjugate Elements J. Group Acting on a Set CHAPTER FOURTEEN: HOMOMORPHISMS EXERCISES A. Examples of Homomorphisms of Finite Groups B. Examples of Homomorphisms of Infinite Groups C. Elementary Properties of Homomorphisms D. Basic Properties of Normal Subgroups E. Further Properties of Normal Subgroups F. Homomorphism and the Order of Elements G. Properties Preserved under Homomorphism H. Inner Direct Products I. Conjugate Subgroups CHAPTER FIFTEEN: QUOTIENT GROUPS EXERCISES A. Examples of Finite Quotient Groups B. Examples of Quotient Groups of R x R C. Relating Properties of H to Properties of G/H D. Properties of G Determined by Properties of G/H and H E. Order of Elements in Quotient Groups F. Quotient of a Group by its Center G. Using the Class Equation to Determine the Size of the Center H. Induction on I G I An Example CHAPTER SIXTEEN: THE FUNDAMENTAL HOMOMORPHISM THEOREM EXERCISES A. Examples of the FHT Applied to Finite Groups B. Example of the FHT Applied to F(R) C. Example of the FHT Applied to Abelian Groups D. Group of Inner Automorphisms of a Group G E. The FHT Applied to Direct Products of Groups F. The First Isomorphism Theorem G. A Sharper Cayley Theorem H. Quotient Groups Isomorphic to the Circle Group I. The Second Isomorphism Theorem J. The Correspondence Theorem K. Cauchy 's Theorem L. Subgroups of p-Groups (Prelude to Sylow) SUPPLEMENTARY PROBLEMS M. p-Sylow Subgroups N. Sylow's Theorem. 0. Lifting Elements from Cosets P. Decomposition of a Finite Abelian Group into p-Groups Q. Basis Theorem for Finite Abelian Groups CHAPTER SEVENTEEN: RINGS: DEFINiTIONS AND ELEMENTARY PROPERTIES EXERCISES A. Examples of Rings B. Ring of Real Functions C. Ring of 2 x 2 Matrices D. Rings of Subsets of a Set E. Ring of Quaternions F. Ring of Endomorphisms G. Direct Product of Rings H. Elementary Properties of Rings I. Properties of Invertible Elements J. Properties of Divisors of Zero K. Boolean Rings L. The Binomial Formula M. Nilpotent and Unipotent Elements CHAPTER EIGHTEEN: IDEALS AND HOMOMORPHISMS EXERCISES A. Examples of Subrings B. Examples of Ideals C. Elementary Properties of Subrings D. Elementary Properties of Ideals E. Examples of Homomorphisms F. Elementary Properties of Homomorphisms G. Examples of Isomorphisms H. Further Properties of Ideals I. Further Properties of Homomorphisms J. A Ring of Endomorphisms CHAPTER NINETEEN: QUOTIENT RINGS EXERCISES A. Examples of Quotient Rings B. Examples of the Use of the FHT C. Quotient Rings and Homomorphic Images in 39R) D. Elementary Applications of the Fundamental Homomorphism Theorem E. Properties of Quotient Rings A/J in Relation to Properties of J F. Prime and Maximal Ideals G. Further Properties of Quotient Rings in Relation to their Ideals H. Zn as a Homomorphic Image of Z CHAPTER TWENTY: INTEGRAL DOMAiNS OPTIONAL EXERCISES A. Characteristic of an Integral Domain B. Characteristic of a Finite Integral Domain C. Finite Rings D. Field of Quotients of an Integral Domain E. Further Properties of the Characteristic of an Integral Domain F. Finite Fields CHAPTER TWENTY-ONE: THE INTEGERS EXERCISES A. Properties of Order Relations in Integral Domains B. Further Properties of Ordered Integral Domains C. Uses of Induction D. Every Integral System Is Isomorphic to / E. Absolute Values F. Problems on the Division Algorithm G. Laws of Multiples H. Principle of Strong Induction CHAPTER TWENTY-TWO: FACTORING INTO PRIMES EXERCISES A. Properties of the Relation "a Divides b" B. Properties of the gcd C. Properties of Relatively Prime Integers D. Further Properties of gcd 's and Relatively Prime Integers E. A Property of the gcd F. Least Common Multiples G. Ideals in 1 H. The gcd and the 1cm as Operations on /. CHAPTER TWENTY-THREE: ELEMENTS OF NUMBER THEORY OPTIONAL EXERCISES A. Solving Single Congruences B. Solving Sets of Congruences C. Elementary Properties of Congruence D. Further Properties of Congruence FL Consequences of Fermat's Theorem F. Consequences of Euler's Theorem G. Wilson's Theorem, and Some Consequences H. Quadratic Residues I. Primitive Roots CHAPTER TWENTY-FOUR: RINGS OF POLYNOMIALS EXERCISES A. Elementary Computation in Domains of Polynomials B. Problems Involving Concepts and Definitions C. Rings Ajxj where A Is Not an Integral Domain D. Domains A(x) where A Has Finite Characteristic E. Subrings and Ideals in A(x) F. Homomorphisms of Domains of Polynomials G. Homomorphisms of Polynomial Domains Induced by a Homomorphism of the Ring of Coefficients H. Polynomials in Several Variables I. Fields of Polynomial Quotients J. Division Algorithm: Uniqueness of Quotient and Remainder CHAPTER TWENTY-FIVE: FACTORING POLYNOMIALS EXERCISES A. Examples of Factoring into Irreducible Factors B. Short Questions Relating to Irreducible Polynomials C. Number of Irreducible Quadratics over a Finite Field D. Ideals in Domains of Polynomials E. Proof of the Unique Factorization Theorem F. A Method for Computing the gcd G. An Automorphism of 194 CHAPTER TWENTY-SIX: SUBSTITUTION IN POLYNOMIALS POLYNOMIALS OVER ZAND Q POLYNOMIALS OVER R AND C EXERCISES A. Finding Roots of Polynomials over Finite Fields B. Finding Roots of Polynomials over Q C. Short Questions Relating to Roots D. Irreducible Polynomials in Q(x) by Eisenstein 's Criterion (and Variations on the Theme) E. Irreducibility of Polynomials of Degree < 4 F. Mapping onto Zn to Determine Irreducibility over Q C. Roots and Factors in A(x) when A Is an Integral Domain H. Polynomial Functions over a Finite Field I. Polynomial Interpolation CHAPTER TWENTY-SEVEN: EXTENSIONS OF FiELDS EXERCISES A. Recognizing Algebraic Elements B. Finding the Minimum Polynomial C. The Structure of Fields F(x)/ D. Short Questions Relating to Field Extensions E. Simple Extensions F. Quadratic Extensions G. Questions Relating to Transcendental Elements H. Common Factors of Two Polynomials: Over F and over Extensions of F I. Derivatives and Their Properties J. Multiple Roots CHAPTER TWENTY-EIGHT: VECTOR SPACES EXERCISES A. Examples of Vector Spaces B. Examples of Subspaces C. Examples of Linear Independence and Bases D. Properties of Subspaces and Bases E. Properties of Linear Transformations F. Isomorphism of Vector Spaces G. Sums of Vector Spaces CHAPTER TWENTY-NINE: DEGREES OF FIELD EXTENSIONS EXERCISES A. Examples of Finite Extensions B. Further Examples of Finite Extensions C. Finite Extensions of Finite Fields D. Degrees of Extensions (Applications of Theorem 2) E. Short Questions Relating to Degrees of Extensions F. Further Properties of Degrees of Extensions G. Fields of Algebraic Elements: Algebraic Numbers CHAPTER THIRTY: RULER AND COMPASS EXERCISES A. Constructible Numbers B. Constructible Points and Constructible Numbers C. Constructible Angles D. Constructible Polygons E. A Constructible Polygon F. A Nonconstructible Polygon G. Further Properties of Constructible Numbers and Figures CHAPTER THIRTY-ONE: GALOIS THEORY: PREAMBLE EXERCISES A. Examples of Root Fields over Q B. Examples of Root Fields over Zp C. Short Questions Relating to Root Fields D. Reducing Iterated Extensions to Simple Extensions E. Roots of Unity and Radical Extensions F. Separable and Inseparable Polynomials G. Multiple Roots over Infinite Fields of Nonzero Characteristic H. An Isomorphism Extension Theorem (Proof of Theorem 3) I. Uniqueness of the Root Field J. Extending Isomorphisms K. Normal Extensions CHAPTER THIRTY-TWO: GALOIS THEORY: THE HEART OF THE MATTER EXERCISES A. Computing a Galois Group B. Computing a Galois Group of Eight Elements C. A Galois Group Equal to S3. D. A Galois Group Equal to D4 E. A Cyclic Galois Group F. A Galois Group Isomorphic to S5 G. Shorter Questions Relating to Automorphisms and Galois Groups H. The Group of Automorphisms of C I. Further Questions Relating to Galois Groups J. Normal Extensions and Normal Subgroups CHAPTER THIRTY-THREE: SOLVING EQUATIONS BY RADICALS EXERCISES A. Finding Radical Extensions B. Solvable Groups C. pth Roots of Elements in a Field D. Another Way of Defining Solvable Groups E. If Gal(K: F) Is Solvable, K Is a Radical Extension of F INDEX