Introduction to abstract algebra

1. Numbers

1.1. Ordering numbers

1.2. The Well-Ordering Principle

1.3. Divisibility

1.4. The Division Algorithm

1.5. Greatest common divisors

1.6. The Euclidean Algorithm

1.7. Primes and irreducibles

1.8. The Fundamental Theorem of Arithmetic

1.9. Exercises

1.10. Study projects

1.11. Notes

2. Functions

2.1. Specifying functions

2.2. Composite functions

2.3. Linear functions

2.4. Semigroups of functions

2.5. Injectivity and surjectivity

2.6. Isomorphisms

2.7. Groups of permutations

2.8. Exercises

2.9. Study projects

2.10. Notes

2.11. Summary

3. Equivalence

3.1. Kernel and equivalence relations

3.2. Equivalence classes

3.3. Rational numbers

3.4. The First Isomorphism Theorem for Sets

3.5. Modular arithmetic

3.6. Exercises

3.7. Study projects

3.8. Notes

4. Groups and Monoids

4.1. Semigroups

4.2. Monoids

4.3. Groups

4.4. Componentwise structure

4.5. Powers

4.6. Submonoids and subgroups

4.7. Cosets

4.8. Multiplication tables

4.9. Exercises

4.10. Study projects

4.11. Notes

5. Homomorphisms

5.1. Homomorphisms

5.2. Normal subgroups

5.3. Quotients

5.4. The First Isomorphism Theorem for Groups

5.5. The Law of Exponents

5.6. Cayley's Theorem

5.7. Exercises

5.8. Study projects

5.9. Notes

6. Rings

6.1. Rings

6.2. Distributivity

6.3. Subrings

6.4. Ring homomorphisms

6.5. Ideals

6.6. Quotient rings

6.7. Polynomial rings

6.8. Substitution

6.9. Exercises

6.10. Study projects

6.11. Notes

7. Fields

7.1. Integral domains

7.2. Degrees

7.3. Fields

7.4. Polynomials over fields

7.5. Principal ideal domains

7.6. Irreducible polynomials

7.7. Lagrange interpolation

7.8. Fields of fractions

7.9. Exercises

7.10. Study projects

7.11. Notes

8. Factorization

8.1. Factorization in integral domains

8.2. Noetherian domains

8.3. Unique factorization domains

8.4. Roots of polynomials

8.5. Splitting fields

8.6. Uniqueness of splitting fields

8.7. Structure of finite fields

8.8. Galois fields

8.9. Exercises

8.10. Study projects

8.11. Notes

9. Modules

9.1. Endomorphisms

9.2. Representing a ring

9.3. Modules

9.4. Submodules

9.5. Direct sums

9.6. Free modules

9.7. Vector spaces

9.8. Abelian groups

9.9. Exercises

9.10. Study projects

9.11. Notes

10. Group Actions

10.1. Actions

10.2. Orbits

10.3. Transitive actions

10.4. Fixed points

10.5. Faithful actions

10.6. Cores

10.7. Alternating groups

10.8. Sylow Theorems

10.9. Exercises

10.10. Study projects

10.11. Notes

11. Quasigroups

11.1. Quasigroups

11.2. Latin squares

11.3. Division

11.4. Quasigroup homomorphisms

11.5. Quasigroup homotopies

11.6. Principal isotopy

11.7. Loops

11.8. Exercises

11.9. Study projects

11.10. Notes

Index