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1 Preliminaries 11 1.1 Well Ordering Principle . 19 1.3 Mathematical Induction . 24 1.4 Functions . 30 2 Basic Group Theory 39 2.1 De nition and Examples . 39 2.2 Basic Properties . 43 3 Subgroups 51 3.1 Examples . 51 3.2 Subgroup Tests . 53 4 Normal Subgroups and Lagrange Theorem 63 4.1 Cosets and their operations . 63 4.2 Normal Subgroup and Factor Group . 66 5 Homomorphisms and Isomorphisms 75 5.1 Homomorphisms . 75 5.2 Isomorphism Theorems . 80 5.3 Isomorphic In nite Groups . 86 6 Cyclic Groups 93 6.1 De nition and Examples . 93 6.2 Properties of Cyclic Groups . 94 6.3 Fundamental Theorem of Cyclic Groups . 98 7 Permutation Groups 107 7.1 De nition and Notations . 107 7.2 Properties of Permutations . 110 7.3 Alternating Groups . 113 7.4 Cayley's Theorem and Group Actions . 117 8 Automorphisms 127 8.1 Automorphisms of Cyclic Groups . 127 8.2 Inner Automorphisms . 129 9 Abelian Groups and Solvable Groups 137 9.1 Abelian Groups with Prime Orders . 137 9.2 Solvable Groups . 139 10 Products 147 10.1 External Direct Products . 147 10.2 Internal Direct Products . 149 11 Sylow Theorems 155 11.1 First Sylow Theorem . 155 11.2 Second Sylow Theorem and Simple Groups . 160 11.3 Third Sylow Theorem and Examples .162 12 Basic Ring Theory 169 12.1 De nition and Examples of Rings . 169 12.2 Polynomial Rings: New Ring From A Ring . 172 12.3 Properties of Ring Elements and Subrings . 175 13 More Rings 183 13.1 Integral Domains . 183 13.2 Fields . 185 13.3 Characteristics of Integral Domains . 189 13.4 Localizations of Integral Domains . 192 14 Polynomial Rings 199 14.1 Addition and Multiplication . 199 14.2 Division and Factors . 201 14.3 Zeros Of A Polynomial . 205 14.4 Formal Power Series Rings . 207 15 Ideals and Factor Rings 211 15.1 Ideals . 211 15.2 Quotient Rings . 214 15.3 Principal Ideal Domain (PID) . 217 15.4 Prime and Maximal Ideals . 220 16 Ring Homomorphisms 231 16.1 De nition . 232 16.2 Homomorphism and its Kernel . 236 16.3 Isomorphism Theorems . 242 17 Factorizations of Polynomials 249 17.1 Reducible and Irreducible Polynomials . 249 17.2 Tests for Irreducibility of Polynomials over Z . 252 17.3 Irreducible Polynomials and Fields . 256 17.4 Prime Numbers and Prime Polynomials . 260 18 Euclidean Domains 269 18.1 De nition and Examples . 269 18.2 Divisions In Rings . 272 19 Factorizations in an Integral Domain 281 19.1 De nitions and Examples . 281 19.2 ED, PID, & UFD . 284 20 Vector Spaces 289 20.1 De nition and Examples . 289 20.2 Subspaces and Spanning Sets . 295 20.3 Basis of a Vector Space . 300 21 Basic Field Theory: Prime Fields 311 21.1 Prime Fields . 311 21.2 Finite and In nite Fields . 313 22 Field Extensions Using a Zero of a Polynomial 319 22.1 Irreducible Polynomials and Extension Fields . 319 22.2 Zero of a Polynomial . 323 22.3 Splitting Field and Field Extensions . 326 22.4 Uniqueness of Splitting Fields . 330 23 Algebraic Extensions 23.1 Finite Extensions . 339 23.2 Minimal Polynomials . 343 23.3 Fundamental Theorem of Algebra . 350 24Geometric Constructions 355 24.1 Constructible Constructions . 355 24.2 Trisection . 358 25 Separable and Inseparable Extensions 363 25.1 Perfect Fields and Multiplicity of a Zero . 363 25.2 Separable Polynomials and Extensions . 365 26 Galois Theory 371 26.1 Normal Extensions . 371 26.2 Galois Extensions . 373 26.3 Fundamental Theorem of Galois Theory . 380 27 Appendix: Matrices and Their Determinants 387 27.1 Matrices . 387 27.2 Determinants . 392 Bibliography 397