Introduction to Abstract Algebra by Samir Siksek

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Introduction to Abstract Algebra written by Samir Siksek . This is an other great mathematics book cover the following topics of Problem Solving.

Chapter I. Prologue
Who Am I? A Jolly Good Read!, Proofs, Acknowledgements and Corrections

Chapter II. FAQ

Chapter III. Algebraic Reorientation
Sets, Binary Operations, Vector Operations, Operations on PolynomialsComposition of Functions, Composition Tables, Commutativity and Associativity, Where are the Proofs?, The Quaternionic Number System (do not read)

Chapter IV. Matrices—Read On Your Own
What are Matrices?, Matrix Operations, Where do matrices come from?, How to think about matrices?, Why Column Vectors?, Multiplicative Identity and Multiplicative Inverse, Rotations

Chapter V. Groups
The Definition of a Group, First Examples (and Non-Examples), Abelian Groups, Symmetries of a Square

Chapter VI. First Theorems
Getting Relaxed about Notation, Additive Notation

Chapter VII. More Examples of Groups
Matrix Groups I, Congruence Classes

Chapter VIII. Orders and Lagrange’s Theorem
The Order of an Element, Lagrange’s Theorem—Version

Chapter IX. Subgroups
What Were They Again?, Criterion for a Subgroup, Roots of Unity, Matrix Groups II, Differential Equations, Non-Trivial and Proper Subgroups, Lagrange’s Theorem—Version 2

Chapter X. Cyclic Groups and Cyclic Subgroups
Lagrange Revisited, Subgroups of Z

Chapter XI. Isomorphisms

Chapter XII. Cosets
Geometric Examples, Solving Equations, Index, The First Innermost Secret of Cosets, The Second Innermost Secret of Cosets, Lagrange Super-Strength

Chapter XIII. Quotient Groups
Congruences Modulo Subgroups, Congruence Classes and Cosets, R/Z, R/Q, Well-Defined and Proofs

Chapter XIV. Symmetric Groups
Motivation, Injections, Surjections and Bijections, The Symmetric Group, A Nice Application of Lagrange’s Theorem, Cycle Notation, Permutations and Transpositions, Even and Odd Permutations

Chapter XV. Rings
Definition, Examples, Subrings, The Unit Group of a Ring, The Unit Group of the Gaussian Integers 117

Chapter XVI. Fields

Chapter XVII. Congruences Revisited
Units in Z/mZ, Fermat’s Little Theorem, Euler’s Theorem, Vale Dicere

Appendices

Appendix A. 2012 Introduction to Abstract Algebra Paper

Appendix B. 2013 Introduction to Abstract Algebra Paper

Appendix C. 2014 Introduction to Abstract Algebra Paper

Appendix D. 2015 Introduction to Abstract Algebra Paper

Appendix E. The Forgotten Joys of Analytic Irresponsibility

E.1. The Mathematical Equivalent of an X-Rated DVD

E.2. Nothing to see here—move along please

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SHORTCUT TRICKS (Division)

Divisible by 2 Shortcut trick

Divisible by 3 Shortcut trick

Divisible by 4 Shortcut trick

Divisible by 5 Shortcut trick

Divisible by 6 Shortcut trick

Divisible by 7 Shortcut trick

Divisible by 8 Shortcut trick

Divisible by 9 Shortcut trick

Divisible by 10 Shortcut trick

SHORTCUT TRICKS (Prime Number)

Find the prime number from 1 to 100 just in 5 second (MATH PRIME NUMBER SHORTCUT TRICK from 1 to 100 number) ?

Find a large number "A" is prime number or not (MATH PRIME NUMBER SHORTCUT TRICK By using Square Root method) ?

Find the composite number from 1 to 100 just in 5 second (MATH COMPOSITE NUMBER SHORTCUT TRICK from 1 to 100 number) ?

Worksheets (Solved)

Matrix Logarithm

Trigonometric Identities

Application of Trigonometry

Integration