# Abstract Algebra Notes
## Introdcution
- This note is for my study of abstract algebra. The purpose to make this note is to record of my personal understanding and help me recall my memory in the easiest way when needed. For simplicity, proofs are excluded in this note, except for those that can enhance understanding of the propositions or theorems.
- The textbook used is [Abstract Algebra by Joseph H. Silverman](https://libgen.is/book/index.php?md5=175D9EC0A4A406DEE33D558868FC8A64), which was once selected as a course book for **Introduction to Algebra** by the Department of Mathematics at National Taiwan University. While this book is written for easy readability, it omits some important propositions and definitions. To gain a deeper understanding of abstract algebra, this note incorporates content from some additional sources.
- This note will be better to read in the dark mode.
## Outline
- ### Groups
- [Part 1 - Chapter2](https://hackmd.io/CHxTUDkfS92xq5Bc9-qPsg)
- 2.1. Introduction to Groups
- 2.2. Abstract Groups
- 2.3. Interesting Examples of Groups
- 2.4. Group Homomorphisms
- 2.5. Subgroups, Cosets, and Lagrange’s Theorem
- 2.6. Products of Groups
- [Part 2 - Chapter6](https://hackmd.io/1KSropAUSmO0eTa1imMT4w)
- 6.1. Normal Subgroups and Quotient Groups
- 6.2. Groups Acting on Sets
- 6.3. The Orbit-Stabilizer Counting Theorem
- 6.4. Sylow’s Theorem
- 6.5. Two Counting Lemmas
- 6.6. Double Cosets and Sylow’s Theorem
- [Part 3 - Chapter12](https://hackmd.io/Kw3GNbh1Q-mUh4jucescXQ)
- 12.1. Permutation Groups
- 12.2. Cayley’s Theorem
- 12.3. Simple Groups
- 12.4. Composition Series
- 12.5. Automorphism Groups
- 12.6. Semidirect Products of Groups
- 12.7. The Structure of Finite Abelian Groups
- ### Rings
- [Part 1 - Chapter3](https://hackmd.io/5izknlFIQwCh7GHp8_xLUw)
- 3.1. Introduction to Rings 63
- 3.2. Abstract Rings and Ring Homomorphisms
- 3.3. Interesting Examples of Rings 65
- 3.4. Some Important Special Types of Rings
- 3.5. Unit Groups and Product Rings
- 3.6. Ideals and Quotient Rings
- 3.7. Prime Ideals and Maximal Ideals
- [Part 2 - Chapter7](https://hackmd.io/vUNEt4KFSvOMm5yZZATseQ)
- 7.1. Irreducible Elements and Unique Factorization Domains
- 7.2. Euclidean Domains and Principal Ideal Domains
- 7.3. Factorization in Principal Ideal Domains
- 7.4. The Chinese Remainder Theorem
- 7.5. Field of Fractions
- 7.6. Multivariate and Symmetric Polynomials
- ### Fields
- [Part 1 - Chpater5](https://hackmd.io/7rwr4TX5RIepD_D0JsuGnA)
- 5.1. Introduction to Fields
- 5.2. Abstract Fields and Homomorphisms
- 5.3. Interesting Examples of Fields
- 5.4. Subfields and Extension Fields
- 5.5. Polynomial Rings
- 5.6. Building Extension Fields
- 5.7. Finite Fields
- [Part 2 - Chapter8](https://hackmd.io/uJXu_vLfSWKV42LuNi3n3Q)
- 8.1. Algebraic Numbers and Transcendental Numbers
- 8.2. Polynomial Roots and Multiplicative Subgroups
- 8.3. Splitting Fields, Separability, and Irreducibility
- 8.4. Finite Fields Revisited 200
- 8.5. Gauss’s Lemma, Eisenstein’s Irreducibility Criterion, and Cyclotomic Polynomials
- 8.6. Ruler and Compass Constructions
- ### Vector Spaces
- [Part 1 - Chapter 4](https://hackmd.io/Nm4EMIFCQbaMgaShpt8rHw)
- 4.1. Introduction to Vector Spaces
- 4.2. Vector Spaces and Linear Transformations
- 4.3. Interesting Examples of Vector Spaces
- 4.4. Bases and Dimension
- [Part 2 - Chapter 10](https://hackmd.io/-pCmJOm2QVu5GA2MVLehNg)
- 10.1. Vector Space Homomorphisms (aka Linear Transformations)
- 10.2. Endomorphisms and Automorphisms
- 10.3. Linear Transformations and Matrices
- 10.4. Subspaces and Quotient Spaces
- 10.5. Eigenvalues and Eigenvectors
- 10.6. Determinants
- 10.7. Determinants, Eigenvalues, and Characteristic Polynomials
- 10.8. Inifinite-Dimensional Vector Spaces
- ### Modules
- [Part 1 - Chapters11](https://hackmd.io/6f-DduC0TaewXRpzzmamCQ)
- 11.1. What Is a Module?
- 11.2. Examples of Modules
- 11.3. Submodules and Quotient Modules
- 11.4. Free Modules and Finitely Generated Modules
- 11.5. Homomorphisms, Endomorphisms, Matrices
- 11.6. Noetherian Rings and Modules
- 11.7. Matrices with Entries in a Euclidean Domain
- 11.8. Finitely Generated Modules over Euclidean Domains
- 11.9. Applications of the Structure Theorem
- [Part 2 - Chapters13](https://hackmd.io/sgNTw-bPSLKKMcNQl5KryQ)
- 13.1. Multilinear Maps and Multilinear Forms
- 13.2. Symmetric and Alternating Forms
- 13.3. Alternating Forms on Free Modules
- 13.4. The Determinant Map
- ### Galois Theory
- [Chapter9](https://hackmd.io/3rFDdAWoQuen6zcvS6V77A)
- 9.1. What Is Galois Theory?
- 9.2. A Quick Review of Polynomials and Field Extensions
- 9.3. Fields of Algebraic Numbers
- 9.4. Algebraically Closed Fields
- 9.5. Automorphisms of Fields
- 9.6. Splitting Fields — Part 1
- 9.7. Splitting Fields — Part 2
- 9.8. The Primitive Element Theorem
- 9.9. Galois Extensions
- 9.10. The Fundamental Theorem of Galois Theory
- 9.12. Galois Theory of Finite Fields
- 9.13. A Plethora of Galois Equivalences
- 9.14. Cyclotomic Fields and Kummer Fields
- 9.15. Application: Insolubility of Polynomial Equations by Radicals
- 9.16. Linear Independence of Field Automorphisms
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