Abstract Algebra Theory and Applications by Thomas W. Judson

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Abstract Algebra Theory and Applications written by Thomas W. Judson cover the following topics.

1. Preliminaries (A Short Note on Proofs, Sets and Equivalence Relations)

2. The Integers
Mathematical Induction, The Division Algorithm

3. Groups
Integer Equivalence Classes and Symmetries, Definitions and Examples, Subgroups

4. Cyclic Groups
Cyclic Subgroups, Multiplicative Group of Complex Numbers, The Method of Repeated Squares

5. Permutation Groups
Definitions and Notation, Dihedral Groups

6. Cosets and Lagrange’s Theorem
Cosets, Lagrange’s Theorem, Fermat’s and Euler’s Theorems

7. Introduction to Cryptography
Private Key Cryptography, Public Key Cryptography

8. Algebraic Coding Theory
Error-Detecting and Correcting Codes, Linear Codes, Parity-Check and Generator Matrices, Efficient Decoding

9. Isomorphisms
Definition and Examples, Direct Products

10. Normal Subgroups and Factor Groups
Factor Groups and Normal Subgroups, The Simplicity of the Alternating Group

11. Homomorphisms
Group Homomorphisms, The Isomorphism Theorems

12. Matrix Groups and Symmetry
Matrix Groups, Symmetry

13. The Structure of Groups
Finite Abelian Groups, Solvable Groups

14. Group Actions
Groups Acting on Sets, The Class Equation, Burnside’s Counting Theorem

15. The Sylow Theorems
The Sylow Theorems, Examples and Applications

16. Rings
Rings, Integral Domains and Fields, Ring Homomorphisms and Ideals, Maximal and Prime Ideals, An Application to Software Design

17. Polynomials
Polynomial Rings, The Division Algorithm, Irreducible Polynomials,

18. Integral Domains
Fields of Fractions, Factorization in Integral Domains

19. Lattices and Boolean Algebras
Lattices, Boolean Algebras, The Algebra of Electrical Circuits

20. Vector Spaces
Definitions and Examples, Subspaces, Linear Independence

21. Fields
Extension Fields, Splitting Fields, Geometric Constructions

22. Finite Fields
Structure of a Finite Field, Polynomial Codes

23. Galois Theory
Field Automorphisms, The Fundamental Theorem, Applications

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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