part 1 Part I: Numbers, Polynomials, and Factoring -- chapter 1 The Natural Numbers -- chapter 2 The Integers -- chapter 3 Modular Arithmetic -- chapter 4 Polynomials with Rational Coefficients -- chapter 5 Factorization of Polynomials -- chapter Section I in a Nutshell -- part Part II: Rings, Domains, and Fields -- chapter 6 Rings -- chapter 7 Subrings and Unity -- chapter 8 Integral Domains and Fields -- chapter 9 Ideals -- chapter 10 Polynomials over a Field -- chapter Section II in a Nutshell -- part 3 Part III: Ring Homomorphisms and Ideals -- chapter 11 Ring Homomorphisms -- chapter 12 The Kernel -- chapter 13 Rings of Cosets -- chapter 14 The Isomorphism Theorem for Rings -- chapter 15 Maximal and Prime Ideals -- chapter 16 The Chinese Remainder Theorem -- chapter Section III in a Nutshell -- part 4 Part IV: Groups -- chapter 17 Symmetries of Geometric Figures -- chapter 18 Permutations -- chapter 19 Abstract Groups -- chapter 20 Subgroups -- chapter 21 Cyclic Groups -- chapter Section IV in a Nutshell -- part 5 Part V: Group Homomorphisms -- chapter 22 Group Homomorphisms -- chapter 23 Structure and Representation -- chapter 24 Cosets and Lagrange’s Theorem -- chapter 25 Groups of Cosets -- chapter 26 The Isomorphism Theorem for Groups -- chapter Section V in a Nutshell -- part 6 Part VI: Topics from Group Theory -- chapter 27 The Alternating Groups -- chapter 28 Sylow Theory: The Preliminaries -- chapter 29 Sylow Theory: The Theorems -- chapter 30 Solvable Groups -- chapter Section VI in a Nutshell -- part 7 Part VII: Unique Factorization -- chapter 31 Quadratic Extensions of the Integers -- chapter 32 Factorization -- chapter 33 Unique Factorization -- chapter 34 Polynomials with Integer Coefficients -- chapter 35 Euclidean Domains -- chapter Section VII in a Nutshell -- part 8 Part VIII: Constructibility Problems -- chapter 36 Constructions with Compass and Straightedge -- chapter 37 Constructibility and Quadratic Field Extensions -- chapter 38 The Impossibility of Certain Constructions -- chapter Section VIII in a Nutshell -- part 9 Part IX: Vector Spaces and Field Extensions -- chapter 39 Vector Spaces I -- chapter 40 Vector Spaces II -- chapter 41 Field Extensions and Kronecker’s Theorem -- chapter 42 Algebraic Field Extensions -- chapter 43 Finite Extensions and Constructibility Revisited -- chapter Section IX in a Nutshell -- part 10 Part X: Galois Theory -- chapter 44 The Splitting Field -- chapter 45 Finite Fields -- chapter 46 Galois Groups -- chapter 47 The Fundamental Theorem of Galois Theory -- chapter 48 Solving Polynomials by Radicals -- chapter Section X in a Nutshell -- chapter Hints and Solutions -- chapter Guide to Notation.