1 Monoids and Groups

1.1 Examples of Groups and MonoidsWhen Is a Monoid a Group?

1.2 Exercises

2 Lagrange’s Theorem, Cosets, and an Application to Number Theory

2.1 Cosets

2.2 Fermat’s Little Theorem

2.3 Exercises

3 Cauchy’s Theorem: Showing that a Number Is Greater Than 1

3.1 The Exponent

3.2 The symmetric group Sn: Our Main Example

3.3 The Product of Two Subgroups

3.4 Exercises

4 Structure of Groups: Homomorphisms, Isomorphisms, and Invariants

4.1 Homomorphic Images

4.2 Exercises

5 Normal Subgroups: The Building Blocks of the Structure Theory

5.1 The Residue Group

5.2 Noether’s Isomorphism Theorems

5.3 Conjugates in Sn

5.4 The Alternating Group

5.5 Exercises

6 Classifying Groups: Cyclic Groups and Direct Products

6.1 Cyclic Groups

6.2 Generators of a Group

6.3 Direct Products

6.4 Application: Some Algebraic Cryptosystems

6.5 Exercises

7 Finite Abelian Groups

7.1 Abelian p-Groups

7.2 Proof of the Fundamental Theorem for Finite abelian Groups

7.3 The Classification of Finite abelian Groups

7.4 Exercises

8 Generators and Relations

8.1 Description of Groups of Low Order

8.3 Exercises

9 When Is a Group a Group? (Cayley’s Theorem)

9.1 The Generalized Cayley Theorem

9.2 Introduction to Group Representations

9.3 Exercises

10 Conjugacy Classes and the Class Equation

10.1 The Center of a Group

10.2 Exercises

11 Sylow Subgroups

11.1 Groups of Order Less Than 60

11.2 Finite Simple Groups

11.3 Exercises

12 Solvable Groups: What Could Be Simpler?

12.1 Commutators

12.2 Solvable Groups

12.3 Automorphisms of Groups

12.4 Exercises

13 Groups of Matrices

13.1 Exercises

14 An Introduction to Rings

14.1 Domains and Skew Fields

14.2 Left Ideals

14.3 Exercises

15 The Structure Theory of Rings

15.1 Ideals

15.2 Noether’s Isomorphism Theorems for Rings

15.3 Exercises

16 The Field of Fractions: A Study in Generalization

16.1 Intermediate Rings

16.2 Exercises

17 Polynomials and Euclidean Domains

17.1 The Ring of Polynomials

17.2 Euclidean Domains

17.3 Unique Factorization

17.4 Exercises

18 Principal Ideal Domains: Induction without Numbers

18.1 Prime Ideals

18.2 Noetherian RingsExercises

19 Roots of Polynomials

19.1 Finite Subgroups of Fields

19.2 Primitive Roots of 1

19.3 Exercises

20 Applications: Famous Results from Number Theory

20.1 A Theorem of Fermat

20.2 Addendum: “Fermat’s Last Theorem”

20.3 Exercises

21 Irreducible Polynomials

21.1 Polynomials over UFDs

21.2 Eisenstein’s Criterion

21.3 Exercises

22 Field Extensions: Creating Roots of Polynomials

22.1 Algebraic Elements

22.2 Finite Field Extensions

22.3 Exercises

23 The Geometric Problems of Antiquity

23.1 Construction by Straight Edge and Compass

23.2 Algebraic Description of Constructibility

23.3 Solution of the Geometric Problems of Antiquity

23.4 Exercises

24 Adjoining Roots to Polynomials: Splitting Fields

24.1 Splitting Fields

24.2 Separable Polynomials and Separable Extensions

24.3 Exercises

25 Finite Fields

25.1 Uniqueness

25.2 Existence

25.3 Exercises

26 The Galois Correspondence

26.1 The Galois Group of a Field Extension

26.2 The Galois Group and Intermediate Fields

26.3 Exercises

27 Applications of the Galois Correspondence

27.1 Finite Separable Field Extensions and the Normal Closure

27.2 The Galois Group of a Polynomial

27.3 Constructible n-gons

27.4 Finite Fields

27.5 The Fundamental Theorem of Algebra

27.6 Exercises

28 Solving Equations by Radicals

28.1 Radical Extensions

28.2 Solvable Galois Groups

28.3 Computing the Galois Group

28.4 Exercises

29 Integral Extensions

29.1 Exercises

30 Group Representations and their Characters

30.1 Exercises

31 Transcendental Numbers: e and π

31.1 Transcendence of e

31.2 Transcendence of π

32 Skew Field Theory

32.1 The Quaternion Algebra

32.2 Polynomials over Skew Fields

32.3 Structure Theorems for Skew Fields

32.4 Exercises

33 Where Do We Go From Here?

33.1 Modules

33.2 Matrix Algebras and their Substructures

33.3 Nonassociative Rings and Algebras

33.4 Hyperfields

33.5 Exercises