First Course in Abstract Algebra, A
Description
This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.
KEY TOPICS: Number Theory: Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic
Congruences; Dates and Days. Groups I: Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups. Commutative Rings I: First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons. Linear Algebra: Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms. Fields: Classical Formulas; Insolvability of the General Quintic; Epilog. Groups II: Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry. Commutative Rings III: Prime Ideals and Maximal Ideals; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases.
MARKET: For all readers interested in abstract algebra.
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Content
Induction
Binomial Coefficients
Greatest Common Divisors
The Fundamental Theorem of Arithmetic
Congruences
Dates and Days
Chapter 2: Groups I
Some Set Theory
Permutations
Groups
Subgroups and Lagrange's Theorem
Homomorphisms
Quotient Groups
Group Actions
Counting with Groups
Chapter 3: Commutative Rings I
First Properties
Fields
Polynomials
Homomorphisms
Greatest Common Divisors
Unique Factorization
Irreducibility
Quotient Rings and Finite Fields
Officers, Magic, Fertilizer, and Horizons
Chapter 4: Linear Algebra
Vector Spaces
Euclidean Constructions
Linear Transformations
Determinants
Codes
Canonical Forms
Chapter 5: Fields
Classical Formulas
Insolvability of the General Quintic
Epilog
Chapter 6: Groups II
Finite Abelian Groups
The Sylow Theorems
Ornamental Symmetry
Chapter 7: Commutative Rings III
Prime Ideals and Maximal Ideals
Unique Factorization
Noetherian Rings
Varieties
Grobner Bases
Hints for Selected Exercises
Bibliography
Index