Algebraic Systems

Author:              Zhiming Ou

                         Mattermatics Learning Center

Publisher:       3265 Public Way

ISBN: 978-1-0677109-3-4

Summary

The study of rules for manipulating formulas and algebraic expressions involving unknowns and real or complex numbers is called elementary algebra. Human also created other systems of algebra. Linear algebra studies vector spaces with a finite dimension; abstract or modern algebra studies groups, rings, and fields; algebraic number theory studies algebraic numbers and transcendental numbers, ideals.

In this book, I summarized the structure and rules of many algebraic systems. Algebraic structures, together with the associated homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures.

Theoretical physics draws on Lie algebras. Crystal structure and gauge symmetry can be described by groups. In mathematics, algebraic topology and algebraic geometry apply algebraic methods extensively.

 

Content

Chapter 1 Groups         5

  • 1 Concepts

1.1 Binary Operations         5

1.2 Definition of Group and Subgroups           6

1.3 Types of groups          7

1.4 Examples       8

  • 2 Structures of Subgroups 13

2.1 Cosets and the Quotient Group        13

2.2 Center, Centralizer and Conjugacy Class       14

2.3 Operation of a Group on a Set       18

2.4 The Sylow Theorems         19

2.5 Constructing New Groups      26

  • 3 Relations between Groups 27

3.1 Isomorphism         27

3.2 Homomorphism      29

  • 4 Special Groups 30

4.1 Finite Simple Groups      30

4.2 The Free Group     31

4.3 Symmetry Groups     33

4.4 Symmetry and Counting     35

Burnside’s Lemma

  • 5 Representations of a Group 35

5.1 Definitions

5.2 Reducibility        37

5.3 Characters       38

 

Chapter 2 Rings         41

  • 2.1 Rings and Integral Domains 41

Elements in a ring        42

Characteristic of a Ring      44

Substructure of a Ring       44

  • 2.2 Ideals 45

Constructions of Rings     46

Quotient Rings

Prime Ideal

Maximal Ideal

Direct Product

  • 2.3 Ring Homomorphisms 50

Properties

Field of Quotients         52

  • 2.4 Polynomial Rings 53

Irreducible Polynomials and Reducible Polynomial        54

Primitive Polynomials

Eisenstein's Criterion

Unique Factorization in Z[x]

  • 2.5 Divisibility in Integral Domain 56

 

Chapter 3 Fields        59

  • 3.1 Adjoining Fields 59

Quadratic Fields     59

Algebraic integers        60

Ideals in OK     60

Ideal Classes       62

  • 3.2 Splitting Fields 63

Fundamental Theorem of Field Theory       63

Existence of Splitting Fields

The Splitting Theorem

Cyclotomic Fields          65

  • 3.3 General Algebraic Extensions 66

Primitive Element Theorem

Degree of an Extension

  • 3.4 Finite Fields 68

Classification of Finite Fields  

Explicit Construction of GF(pn)        69

Structure of a Finite Field         71

Discrete logarithm         72

Roots of unity       73

Number of monic irreducible polynomials over a finite field          74

 

Chapter 4 Galois Theory       75

  • 4.1 Classical Formulas 75
  • 4.2 Solvable by Radicals 76
  • 4.3 The Galois Group 78

Properties of Automorphisms         79

Fundamental Theorem of Galois Theory       82

  • 4.4 Solvable Groups 82

Properties

  • 4.5 The Galois Theorem 87

Insolvability of a Quintic Equation         88

 

Chapter 5 Quaternions       89

  • 5.1 Concepts and Operations 90

Hamilton product

Scalar and Vector Parts

Conjugation, Norm, and Reciprocal

Square roots of −1

  • 5.2 Quaternions and the Geometry of R3 93

Rotation in Space

  • 5.3 Functions of a quaternion variable 94