Mathematics document containing theorems and formulas.

TABLE OF CONTENTS CHAPTER 0 PREREQUISITES 0. 1 Elementary Number Theory 0. 2 Set Theory 0. 3 Linear Algebra CHAPTER 1 GROUP FUNDAMENTALS 1. 1 Groups and Subgroups 1. 2 Permutation Groups 1. 3 Cosets, Normal Subgroups and Homomorphisms 1. 4 The Isomorphism Theorems 1. 5 Direct Products CHAPTER 2 RING FUNDAMENTALS 2. 1 Basic Definitions and Properties 2. 2 Ideals, Homomorphisms and Quotient Rings 2. 3 The Isomorphism Theorems For Rings 2. 4 Maximal and Prime Ideals 2. 5 Polynomial Rings 2. 6 Unique Factorization 2. 7 Principal Ideal Domains and Euclidean Domains 2. 8 Rings of Fractions 2. 9 Irreducible Polynomials CHAPTER 3 FIELD FUNDAMENTALS 3. 1 Field Extensions 3. 2 Splitting Fields 3. 3 Algebraic Closures 3. 4 Separability 3. 5 Normal Extensions CHAPTER 4 MODULE FUNDAMENTALS 4. 1 Modules and Algebras 4. 2 The Isomorphism Theorems For Modules 4. 3 Direct Sums and Free Modules 4. 4 Homomorphisms and Matrices 4. 5 Smith Normal Form 4. 6 Fundamental Structure Theorems 4. 7 Exact Sequences and Diagram Chasing CHAPTER 5 SOME BASIC TECHNIQUES OF GROUP THEORY 5. 1 Groups Acting on Sets 5. 2 The Orbit-Stabilizer Theorem 5. 3 Applications to Combinatorics 5. 4 The Sylow Theorems 5. 5 Applications of the Sylow Theorems 5. 6 Composition Series 5. 7 Solvable and Nilpotent Groups5. 8 Generators and Relations CHAPTER 6 GALOIS THEORY 6. 1 Fixed Fields and Galois Groups 6. 2 The Fundamental Theorem 6. 3 Computing a Galois Group Directly 6. 4 Finite Fields 6. 5 Cyclotomic Fields 6. 6 The Galois Group of a Cubic 6. 7 Cyclic and Kummer Extensions 6. 8 Solvability by Radicals 6. 9 Transcendental Extensions Appendix to Chapter 6

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