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.