## Descriptions

#### TABLE OF CONTENT

Chapter 1

Why Abstract Algebra?

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

Abstraction in Algebra.

Chapter 2 Operations

Operations on a Set. Properties of Operations.

Chapter 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.

Chapter 4

Elementary Properties of Groups

Uniqueness of Identity and Inverses. Properties of Inverses.

Direct Product of Groups.

Chapter 5

Subgroups

Definition of Subgroup. Generators and Defining Relations.

Cay ley Diagrams. Center of a Group. Group Codes; Hamming Code.

Chapter 6

Functions

Injective, Surjective, Bijective Function. Composite and Inverse of Functions.

Finite-State Machines. Automata and Their Semigroups.

Chapter 7

Groups of Permutations

Symmetric Groups. Dihedral Groups.

An Application of Groups to Anthropology.

Chapter 8

Permutations of a Finite Set

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

Alternating Groups.

Chapter 9

Isomorphism

The Concept of Isomorphism in Mathematics. Isomorphic and Nonisomorphic Groups.

Cayley’s Theorem.

Group Automorphisms.

Chapter 10 Order of Group Elements

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

Elements.

Chapter 11

Cyclic Groups

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

Groups.

Chapter 12

Partitions and Equivalence Relations

Chapter 13

Counting Cosets

Lagrange’s Theorem and Elementary Consequences.

Survey of Groups of Order ≤ 10.

Number of Conjugate Elements. Group Acting on a Set.

Chapter 14

Homomorphisms

Elementary Properties of Homomorphisms. Normal Subgroups. Kernel and Range.

Inner Direct Products. Conjugate Subgroups.

Chapter 15

Quotient Groups

Quotient Group Construction. Examples and Applications.

The Class Equation. Induction on the Order of a Group.

Chapter 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.

Chapter 17

Rings: Definitions and Elementary Properties

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

Chapter 18

and Homomorphisms

Chapter 19

Quotient Rings

Construction of Quotient Rings. Examples. Fundamental Homomorphism Theorem and

Some Consequences. Properties of Prime and Maximal Ideals.

Chapter 20 Integral Domains

Characteristic of an Integral Domain. Properties of the Characteristic. Finite Fields.

Construction of the Field of Quotients.

Chapter 21

The Integers

Ordered Integral Domains. Well-ordering. Characterization of Up to Isomorphism.

Mathematical Induction. Division Algorithm.

Chapter 22

Factoring into Primes

Ideals of . Properties of the GCD. Relatively Prime Integers. Primes. Euclid’s Lemma.

Unique Factorization.

Chapter 23

of Number Theory (Optional)

Properties of Congruence. Theorems of Fermât and Euler. Solutions of Linear Congruences.

Chinese Remainder Theorem.

Wilson’s Theorem and Consequences. Quadratic Residues. The Legendre Symbol.

Primitive Roots.

Chapter 24

Rings of Polynomials

Motivation and Definitions. Domain of Polynomials over a Field. Division Algorithm.

Polynomials in Several Variables. Fields of Polynomial Quotients.

Chapter 25

Factoring Polynomials

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

Euclidean Algorithm.

Chapter 26

Substitution in Polynomials

Roots and Factors. Polynomial Functions. Polynomials over . Eisenstein’s Irreducibility

Criterion. Polynomials over the Reals. Polynomial Interpolation.

Chapter 27

of Fields

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

Field Extensions.

Chapter 28

Vector Spaces

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

Transformations.

Chapter 29

Degrees of Field Extensions

Simple and Iterated Extensions. Degree of an Iterated Extension.

Fields of Algebraic Elements. Algebraic Numbers. Algebraic Closure.

Chapter 30 Ruler and Compass

Constructible Points and Numbers. Impossible Constructions.

Constructible Angles and Polygons.

Chapter 31

Galois Theory: Preamble

Multiple Roots. Root Field. Extension of a Field. Isomorphism.

Roots of Unity. Separable Polynomials. Normal Extensions.

Chapter 32

Theory: The Heart of the Matter

Field Automorphisms. The Galois Group. The Galois Correspondence. Fundamental

Theorem of Galois Theory.

Computing Galois Groups.

Chapter 33

Solving Equations by Radicals

Radical Extensions. Abelian Extensions. Solvable Groups. Insolvability of the Quin tic.

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