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Rings, Fields and Groups: An Introduction to Abstract Algebra

R. B. J. T. Allenby
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1 Why Abstract Algebra?

History of Algebra. New Algebras. Algebraic Structures. Axioms and Axiomatic Algebra. Abstraction in Algebra.

2 Operations

Operations on a Set. Properties of Operations.

3 The Definition of Groups

Groups. Examples of Infinite and Finite Groups. Examples of Abelian and Nonabelian Groups. Group Tables.
Theory of Coding: Maximum-Likelihood Decoding.

4 Elementary Properties of Groups

Uniqueness of Identity and Inverses. Properties of Inverses.
Direct Product of Groups.

5 Subgroups

Definition of Subgroup. Generators and Defining Relations.
Cayley Diagrams. Center of a Group. Group Codes; Hamming Code.

6 Functions

Injective, Subjective, Bijective Function. Composite and Inverse of Functions.
Finite-State Machines. Automata and Their Semigroups.

7 Groups of Permutations

Symmetric Groups. Dihedral Groups.
An Application of Groups to Anthropology.

8 Permutations of a Finite Set

Decomposition of Permutations into Cycles. Transpositions. Even and Odd Permutations. Alternating Groups.

9 Isomorphism

The Concept of Isomorphism in Mathematics. Isomorphic and Nonisomorphic Groups. Cayley's Theorem.
Group Automorphisms.

10 Order of Group Elements

Powers/Multiples of Group Elements. Laws of Exponents. Properties of the Order of Group Elements.

11 Cyclic Groups

Finite and Infinite Cyclic Groups. Isomorphism of Cyclic Groups. Subgroups of Cyclic Groups.

12 Partitions and Equivalence Relations

13 Counting Cosets

Lagrange's Theorem and Elementary Consequences.
Survey of Groups of Order ≤10. Number of Conjugate Elements. Group Acting on a Set.

14 Homomorphisms

Elementary Properties of Homomorphisms. Normal Subgroups. Kernel and Range.
Inner Direct Products. Conjugate Subgroups.

15 Quotient Groups

Quotient Group Construction. Examples and Applications.
The Class Equation. Induction on the Order of a Group.

16 The Fundamental Homomorphism Theorem

Fundamental Homomorphism Theorem and Some Consequences.
The Isomorphism Theorems. The Correspondence Theorem. Cauchy's Theorem. Sylow Subgroups. Sylow's Theorem. Decomposition Theorem for Finite Abelian Groups.

17 Rings: Definitions and Elementary Properties

Commutative Rings. Unity. Invertibles and Zero-Divisors. Integral Domain. Field.

18 Ideals and Homomorphisms

19 Quotient Rings

Construction of Quotient Rings. Examples. Fundamental Homomorphism Theorem and Some Consequences. Properties of Prime and Maximal Ideas.

20 Integral Domains

Characteristic of an Integral Domain. Properties of the Characteristic. Finite Fields. Construction of the Field of Quotients.

21 The Integers

Ordered Integral Domains. Well-ordering. Characterization of Ζ Up to Isomorphism. Mathematical Induction. Division Algorithm.

22 Factoring into Primes

Ideals of Ζ. Properties of the GCD. Relatively Prime Integers. Primes. Euclid's Lemma. Unique Factorization.

23 Elements of Number Theory (Optional)

Properties of Congruence. Theorems of Fermat and Euler. Solutions of Linear Congruences. Chinese Remainder Theorem.
Wilson's Theorem and Consequences. Quadratic Residues. The Legendre Symbol. Primitive Roots.

24 Rings of Polynomials

Motivation and Definitions. Domain of Polynomials over a Field. Division Algorithm.
Polynomials in Several Variables. Fields of Polynomial Quotients.

25 Factoring Polynomials

Ideals of F[x]. Properties of the GCD. Irreducible Polynomials. Unique factorization.
Euclidean Algorithm.

26 Substitution in Polynomials

Roots and Factors. Polynomial Functions. Polynomials over Q. Eisenstein's Irreducibility Criterion. Polynomials over the Reals. Polynomial Interpolation.

27 Extensions of Fields

Algebraic and Transcendental Elements. The Minimum Polynomial. Basic Theorem on Field Extensions.

28 Vector Spaces

Elementary Properties of Vector Spaces. Linear Independence. Basis. Dimension. Linear Transformations.

29 Degrees of Field Extensions

Simple and Iterated Extensions. Degree of an Iterated Extension.
Fields of Algebraic Elements. Algebraic Numbers. Algebraic Closure.

30 Ruler and Compass

Constructible Points and Numbers. Impossible Constructions.
Constructible Angles and Polygons.

31 Galois Theory: Preamble

Multiple Roots. Root Field. Extension of a Field. Isomorphism.
Roots of Unity. Separable Polynomials. Normal Extensions.

32 Galois Theory: The Heart of the Matter

Field Automorphisms. The Galois Group. The Galois Correspondence. Fundamental Theorem of Galois Theory.
Computing Galois Groups.

33 Solving Equations by Radicals

Radical Extensions. Abelian Extensions. Solvable Groups. Insolvability of the Quintic.

Appendix A: Review of Set Theory

Appendix B: Review of the Integers

Appendix C. Review of Mathematical Induction

Answers Selected Exercises