Preface to the Fourth Edition 1 The Integers 1 1.1 Commutative Rings; Integral Domains 1 1.2 Elementary Properties of Commutative Rings 3 1.3 Ordered Domains 8 1.4 Well-Ordering Principle 11 1.5 Finite Induction; Laws of Exponents 12 1.6 ivisibility 16 1.7 The Euclidean Algorithm 18 1.8 Fundamental Theorem of Arithmetic 23 1.9 Congruences 25 1.10 The Rings Zn 29 1.11 Sets, Functions, and Relations 32 1.12 Isomorphisms and Automorphisms 35 2 Rational Numbers and Fields 38 2.1 Definition of a Field 38 2.2 Construction of the Rationals 42 2.3 Simultaneous Linear Equations 47 2.4 Ordered Fields 52 2.5 Postulates for the Positive Integers 54 2.6 Peano Postulates 57 3 Polynomials 61 3.1 Polynomial Forms 61 3.2 Polynomial Functions 65 3.3 Homomorphisms of Commutative Rings 69 3.4 Polynomials in Several Variables 72 3.5 The Division Algorithm 74 3.6 Units and Associates 76 3.7 Irreducible Polynomials 78 3.8 Unique Factorization Theorem 80 3.9 Other Domains with Unique Factorization 84 3.10 Eisenstein's Irreducibility Criterion 88 3.11 Partial Fractions 90 4 Real Numbers 94 4.1 Dilemma of Pythagoras 94 4.2 Upper and Lower Bounds 96 4.3 Postulates for Real Numbers 98 4.4 Roots of Polynomial Equations 101 4.5 Dedekind Cuts 104 5 Complex Numbers 107 5.1 Definition 107 5.2 The Complex Plane 110 5.3 Fundamental Theorem of Algebra 113 5.4 Conjugate Numbers and Real Polynomials 117 5.5 Quadratic and Cubic Equations 118 5.6 Solution of Quartic by Radicals 121 5.7 Equations of Stable Type 122 6 Groups 124 6.1 Symmetries of the Square 124 6.2 Groups of Transformations 126 6.3 Further Examples 131 6.4 Abstract Groups 133 6.5 Isomorphism 137 6.6 Cyclic Groups 140 6.7 Subgroups 143 6.8 Lagrange's Theorem 146 6.9 Permutation Groups 150 6.10 Even and Odd Permutations 153 6.11 Homomorphisms 155 6.12 Automorphisms; Conjugate Elements 157 6.13 Quotient Groups 161 6.14 Equivalence and Congruence Relations 164 7 Vectors and Vector Spaces168 7.1 Vectors in a Plane 168 7.2 Generalizations 169 7.3 Vector Spaces and Subspaces 171 7.4 Linear Independence and Dimension 176 7.5 Matrices and Row-equivalence 180 7.6 Tests for Linear Dependence 183 7.7 Vector Equations; Homogeneous Equations 188 7.8 Bases and Coordinate Systems 193 7.9 Inner Products 198 7.10 Euclidean Vector Spaces 200 7.11 Normal Orthogonal Bases 203 7.12 Quotient-spaces 206 7.13 Linear Functions and Dual Spaces 208 8 The Algebra of Matrices 214 8.1 Linear Transformations and Matrices 214 8.2 Matrix Addition 220 8.3 Matrix Multiplication 222 8.4 Diagonal, Permutation, and Triangular Matrices 228 8.5 Rectangular Matrices 230 8.6 Inverses 235 8.7 Rank and Nullity 241 8.8 Elementary Matrices 243 8.9 Equivalence and Canonical Form 248 8.10 Bilinear Functions and Tensor Products 251 8.11 Quaternions 255 9 Linear Groups 260 9.1 Change of Basis 260 9.2 Similar Matrices and Eigenvectors 263 9.3 The Full Linear and Affine Groups 268 9.4 The Orthogonal and Euclidean Groups 272 9.5 Invariants and Canonical Forms 277 9.6 Linear and Bilinear Forms 280 9.7 Quadratic Forms 283 9.8 Quadratic Forms Under the Full Linear Group 286 9.9 Real Quadratic Forms Under the Full Linear Group 288 9.10 Quadratic Forms Under the Orthogonal Group 292 9.11 Quadrics Under the Affine and Euclidean Groups 296 9.12 Unitary and Hermitian Matrices 300 9.13 Affine Geometry 305 9.14 Projective Geometry 312 10 Determinants and Canonical Forms 318 10.1 Definition and Elementary Properties of Determinants 318 10.2 Products of Determinants 323 10.3 Determinants as Volumes 327 10.4 The Characteristic Polynomial 331 10.5 The Minimal Polynomial 336 10.6 Cayley-Hamilton Theorem 340 10.7 Invariant Subspaces and Reducibility 342 10.8 First Decomposition Theorem 346 10.9 Second Decomposition Theorem 349 10.10 Rational and Jordan Canonical Forms 352 11 Boolean Algebras and Lattices 357 11.1 Basic Definition 357 11.2 Laws: Analogy with Arithmetic 359 11.3 Boolean Algebra 361 11.4 Deduction of Other Basic Laws 364 11.5 Canonical Forms of Boolean Polynomials 368 11.6 Partial Orderings 371 11.7 Lattices 374 11.8 Representation by Sets 377 12 Transfinite Arithmetic 381 12.1 Numbers and Sets 381 12.2 Countable Sets 383 12.3 Other Cardinal Numbers 386 12.4 Addition and Multiplication of Cardinals 390 12.5 Exponentiation 392 13 Rings and Ideals 395 13.1 Rings 395 13.2 Homomorphisms 399 13.3 Quotient-rings 403 13.4 Algebra of Ideals 407 13.5 Polynomial Ideals 410 13.6 Ideals in Linear Algebras 413 13.7 The Characteristic of a Ring 415 13.8 Characteristics of Fields 418 14 Algebraic Number Fields420 14.1 Algebraic and Transcendental Extensions 420 14.2 Elements Algebraic over a Field 423 14.3 Adjunction of Roots 425 14.4 Degrees and Finite Extensions 429 14.5 Iterated Algebraic Extensions 431 14.6 Algebraic Numbers 435 14.7 Gaussian Integers 439 14.8 Algebraic Integers 443 14.9 Sums and Products of Integers 445 14.10 Factorization of Quadratic Integers 448 15 Galois Theory 452 15.1 Root Fields for Equations 452 15.2 Uniqueness Theorem 454 15.3 Finite Fields 456 15.4 The Galois Group 459 15.5 Separable and Inseparable Polynomials 464 15.6 Properties of the Galois Group 467 15.7 Subgroups and Subfields 471 15.8 Irreducible Cubic Equations 474 15.9 Insolvability of Quintic Equations 478 Bibliography 483 List of Special Symbols 486 Index489
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