ALGEBRA AND ALGEBRAIC COMPUTING ELEMENTS OF. John D. Lipson. Addison-Wesley Publishing Company, Inc.

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1 ELEMENTS OF ALGEBRA AND ALGEBRAIC COMPUTING John D. Lipson University of Toronto PRO Addison-Wesley Publishing Company, Inc. Redwood City, California Menlo Park, California Reading, Massachusetts Amsterdam Don Mills, Ontario Mexico City Sydney Bonn Madrid Singapore Tokyo Bogota Santiago San Juan Wokingham, United Kingdom

2 CONTENTS Preface xin PART ONE Chapter I MATHEMATICAL FOUNDATIONS: SETS AND INTEGERS SETS, RELATIONS, AND FUNCTIONS Terminology of Sets 1.1 Set Membership 1.2 Equality and Inclusion Operations on Sets 2.1 Subset Specification; the Power Set 2.2 Union, Intersection, and Complement 2.3 The Cartesian Product Relations 3.1 The Relation Concept 3.2 Equivalence Relations 3.3 Partial Orders Functions 4.1 The Function Concept 4.2 Composition and Invertibility 4.3 Characteristic Functions 4.4 Functions and Equivalence Relations Notes Vit

3 vlll CONTENTS Chapter II THE INTEGERS 1. Basic Properties 2. Induction and Recursion 3. Division and Divisibility 3.1 Equivalence and Remainders mod m 3.2 Greatest Common Divisors 3.3 Factorization into Primes PART TWO ALGEBRAIC SYSTEMS Prologue to Algebra: Algebralc Systems and Abstraction Chapter IM Chapter IV 1. Levels of Abstraction 2. Heterogeneous Algebras Prologue Notes SEMIGROUPS, MONOIDS, AND GROUPS 1. Basic Definitions and Examples 1.0 Groupoids 1.1 Semigroups 1.2 Monoids 1.3 Groups 2. Basic Properties of Binary Algebraic Systems 2.1 Identity Elements in Semigroups 2.2 Inverses in Monoids and Groups 2.3 Solving Equations over Groups 2.4 Products and Powers in Semigroups and Groups 3. Subalgebras (A Universal Algebra Concept) 3.1 Definition and Examples 3.2 Subalgebras Generated by Subsets 3.3 Subgroups 3.4 Cyclic Groups; Order of Group Elements 4. Morphisms (Another Universal Algebra Concept) 4.1 The Morphism Concept 4.2 Structure Preserving Properties of Morphisms RINGS, INTEGRAL DOMAINS, AND FIELDS 1. The Ring Concept 1.1 Basic Definitions and Examples 1.2 Subrings 1.3 Morphisms of Rings

4 CONTENTS Ix 2. Integral Domains and Fields Zerodivisors and Units; Integral Domains and Fields Field of Quotients Polynomials and Formal Power Series Algebra of Polynomials and Formal Power Series The Division Property of Polynomials Polynomials as Functions Divisibility; Euclidean Domains Divisibility Concepts in Integral Domains Euclidean Domains Chapter V QUOTIENT ALGEBRAS Universal Quotient Algebras Congruence Relations The Quotient Algebra/Morphism Theorems of Universal Algebra Quotient Rings Ideals and Quotient Rings Isomorphism Theorem for Rings Further Theory of Ideals Principal Ideal Domains Unital Subrings; Prime Subfields Prime and Maximal Ideals Notes 170 Chapter VI ELEMENTS OF FIELD THEORY 1. Extension Fields 1.1 The Root Adjunction Problem 1.2 Analysis of Simple Extension Fields 2. The Multiplicative Group of a Finite Field 2.1 Cyclic Property of Finite Fields 2.2 Finite Fields as Algebraic Extensions 3. Uniqueness and Existence of Finite Fields 3.1 Uniqueness of GF( p") 3.2 Existence of GF(/>") Notes

5 [ CONTENTS PARTTHREE ALGEBRAIC COMPUTING 191 Chapter VII ARITHMETIC IN EUCLIDEAN DOMAINS Complexity of Integer and Polynomial Arithmetic Polynomial Arithmetic Integer Arithmetic Computation of Greatest Common Divisors Derivation of Euclid's Algorithm Analysis of Euclid's Algorithm over Z and F[x] Euclid's Extended Algorithm Computation of mod m Inverses Theory of mod m Inverses Computation of mod m Inverses 214 Appendix: The Invariant Relation Theorem Notes 228 Chapter VIII COMPUTATION BY HOMOMORPHIC IMAGES 233 Chapter IX Overview Computation by a Single Homomorphic Image ß-Expressions and their Evaluation Solutions to an Integer Congruence The Homomorphic Image Scheme for Z Chinese Remainder and Interpolation Algorithms A CRA for Euclidean Domains A CRA for Z A CRA for F[x]: Interpolation Computation by Multiple Homomorphic Images The MHI Scheme for Z The MHI Scheme for F[x] The MHI Scheme for Z[x] 277 Appendix 1: Computing Lists of Primes 280 Appendix 2: "Adjoint Solution" to Ax = b Notes 290 THE FAST FOURIER TRANSFORM: ITS ROLE IN COMPUTER ALGEBRA What is the Fast Fourier Transform? The Forward Transform: Fast Multipoint Evaluation The Inverse Transform: Fast Interpolation Feasibility of mod p FFTs 303

6 CONTENTS xl 2. Fast Algorithms for Multiplying Polynomials and Integers Fast Polynomial Multiplication Fast Integer Multiplication Fast Algorithms for Manipulating Formal Power Series Truncated Power Series Revisited Fast Power Series Inversion; Newton's Method Polynomial Root-Finding over Power Series Domains Notes 327 SELECTED BIBLIOGRAPHY 330 INDEX TO NOTATION 333 INDEX 337

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