LECTURE NOTES OF WILLIAM CHEN
INTRODUCTION TO ABSTRACT ALGEBRAIC STRUCTURES
This set of notes has been organized in such a way to create a single volume suitable for a very brief introduction to the theory groups, rings and fields, as well as their applications to counting and coding.
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Chapter 1 : GROUPS >>
- Formal Definition
- Elementary Properties
- Subgroups
- Special Subgroups
Chapter 2 : FURTHER PROPERTIES OF GROUPS >>
- Order
- Lagrange's Theorem
- Cyclic Groups
Chapter 3 : FURTHER EXAMPLES OF GROUPS >>
- The Groups Zn*
- Permutation Groups
- Dihedral Groups
Chapter 4 : GROUP HOMOMORPHISMS AND ISOMORPHISMS >>
- Formal Definition
- Some Properties of Homomorphisms
- Normal Subgroups
- Cosets and Factor Groups
- The Fundamental Theorem of Group Homomorphisms
Chapter 5 : FURTHER TOPICS ON GROUPS >>
- Direct Product of Groups
- Geometric Interpretation of Matrix Groups
Chapter 6 : RINGS >>
- Formal Definition
- Elementary Properties
- Subrings
- Further Properties
Chapter 7 : RING HOMOMORPHISMS AND IDEALS >>
- Ring Homomorphisms
- Ideals
- Quotient Rings
- Fundamental Theorem of Ring Homomorphisms
- Prime and Maximal Ideals
Chapter 8 : POLYNOMIAL RINGS >>
- Introduction and Elementary Properties
- Factorization Properties
Chapter 9 : FIELD EXTENSIONS >>
- Ideals in Polynomial Rings
- The Structure of an Extension Field
- Characteristic of a Field
- Finite Fields
- Connection with Group Theory
Chapter 10 : UNIQUE FACTORIZATION >>
- A Simple Case
- Principal Ideal Domains
- The Simple Case Again
- Euclidean Domains
- A Shortcut
- Two Remarks
Chapter 11 : APPLICATION TO COUNTING >>
- Group Actions
- Burnside's Theorem
- Conjugates
- The Class Equation
Chapter 12 : APPLICATION TO CODING >>
- Introduction
- Group Codes
- Matrix Codes
- Error Detection and Correction
- Decoding in Matrix Codes
- Coset Leaders
- Hamming Codes
- Polynomial Codes
- Connection with Field Theory