This Guide offers a concise overview of the theory of groups, rings, and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are often omitted in the standard graduate course, including linear groups, group representations, the structure of Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, without proofs but often with a discussion of the intuitive ideas behind those proofs. Those looking for a way to review and refresh their basic algebra will benefit from reading this Guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work.
Author(s): Fernando Q. Gouvêa
Series: Dolciani Mathematical Expositions 48
Publisher: Mathematical Association of America
Year: 2012
Language: English
Pages: C, xviii, 309
over
S Title
(C) 2012 byThe Mathematical Association of America
Print Edition ISBN 978-0-88385-355-9
Electronic Edition ISBN 978-1-61444-211-0
Library of Congress Catalog Card Number 2012950687
A Guide to Groups, Rings, and Fields
Editors
List of Published Books
Contents
Preface
A Guide to this Guide
CHAPTER 1 Algebra: Classical, Modern, and Ultramodern
The Beginnings of Modern Algebra
Modern Algebra
Ultramodern Algebra
What Next?
CHAPTER 2 Categories
Categories
Functors
Natural Transformations
Products, Coproducts, and Generalizations
CHAPTER 3 Algebraic Structures
Structures with One Operation
Rings
Actions
Semirings
Algebras
Ordered Structures
CHAPTER 4 Groups and their Representations
Definitions
Groups and homomorphisms
Subgroups
Actions
G acting on itself
Some Important Examples
Permutation groups
Symmetry groups
Other examples
Topological groups
Free groups
Reframing the Definitions
Orbits and Stabilizers
Stabilizers
Orbits
Acting by multiplication
Cosets
Counting cosets and elements
Double cosets
A nice example
Homomorphisms and Subgroups
Kernel, image, quotient
Homomorphism theorems
Exact sequences
Hölder's dream
Many Cheerful Subgroups
Generators, cyclic groups
Elements of finite order
Finitely generated groups and the Burnside problem
Other nice subgroups
Conjugation and the class equation
p-groups
Sylow's Theorem and Sylow subgroups
Sequences of Subgroups
Composition series
Central series, derived series, nilpotent, solvable
New Groups from Old
Direct products
Semidirect products
Isometries of R3
Free products
Direct sums of abelian groups
Inverse limits and direct limits
Generators and Relations
Definition and examples
Cayley graphs
The word problem
Abelian Groups
Torsion
The structure theorem
Small Groups
Order four, order p2
Order six, order pq
Order eight, order p3
And so on
Groups of Permutations
Cycle notation and cycle structure
Conjugation and cycle structure
Transpositions as generators
Signs and the alternating groups
Transitive subgroups
Automorphism group of S_n
Some Linear Groups
Definitions and examples
Generators
The regular representation
Diagonal and upper triangular
Normal subgroups
PGL
Linear groups over finite fields
Representations of Finite Groups
Definitions
Examples
Constructions
Decomposing into irreducibles
Direct products
Characters
Character tables
Going through quotients
Going up and down
Representations of S_4
CHAPTER 5 Rings and Modules
Definitions
Rings
Modules
Torsion
Bimodules
Ideals
Restriction of scalars
Rings with few ideals
More Examples, More Definitions
The integers
Fields and skew fields
Polynomials
R-algebras
Matrix rings
Group algebras
Monsters
Some examples of modules
Nil and nilpotent ideals
Vector spaces as K[X]-modules
Q and Q/Z as Z-modules
Why study modules?
Homomorphisms, Submodules, and Ideals
Submodules and quotients
Quotient rings
Irreducible modules, simple rings
Small and large submodules
Composing and Decomposing
Direct sums and products
Complements
Direct and inverse limits
Products of rings
Free Modules
Definitions and examples
Vector spaces
Traps
Generators and free modules
Homomorphisms of free modules
Invariant basis number
Commutative Rings, Take One
Prime ideals
Primary ideals
The Chinese Remainder Theorem
Rings of Polynomials
Degree
The evaluation homomorphism
Integrality
Unique factorization and ideals
Derivatives
Symmetric polynomials and functions
Polynomials as functions
The Radical, Local Rings, and Nakayama's Lemma
The Jacobson radical
Local rings
Nakayama's Lemma
Commutative Rings: Localization
Localization
Fields of fractions
An important example
Modules under localization
Ideals under localization
Integrality under localization
Localization at a prime ideal
What if R is not commutative?
Hom
Making Hom a module
Functoriality
Additivity
Exactness
Tensor Products
Definition and examples
Examples
Additivity and exactness
Over which ring?
When R is commutative
Extension of scalars, aka base change
Extension and restriction
Tensor products and Hom
Finite free modules
Tensoring a module with itself
Tensoring two rings
Projective, Injective, Flat
Projective modules
Injective modules
Flat modules
If R is commutative
Finiteness Conditions for Modules
Finitely generated and finitely cogenerated
Artinian and Noetherian
Finite length
Semisimple Modules
Definitions
Basic properties
Socle and radical
Finiteness conditions
Structure of Rings
Finiteness conditions for rings
Simple Artinian rings
Semisimple rings
Artinian rings
Non-Artinian rings
Factorization in Domains
Units, irreducibles, and the rest
Existence of factorization
Uniqueness of factorization
Principal ideal domains
Euclidean domains
Greatest common divisor
Dedekind domains
Finitely Generated Modules over Dedekind Domains
The structure theorems
Application to abelian groups
Application to linear transformations
CHAPTER 6 Fields and Skew Fields
Fields and Algebras
Some examples
Characteristic and prime fields
K-algebras and extensions
Two kinds of K-homomorphisms
Generating sets
Compositum
Linear disjointness
Algebraic Extensions
Definitions
Transitivity
Working without an A
Norm and trace
Algebraic elements and homomorphisms
Splitting fields
Algebraic closure
Finite Fields
Transcendental Extensions
Transcendence basis
Geometric examples
Noether Normalization
Luroth's Theorem
Symmetric functions
Separability
Separable and inseparable polynomials
Separable extensions
Separability and tensor products
Norm and trace
Purely inseparable extensions
Separable closure
Primitive elements
Automorphisms and Normal Extensions
Automorphisms
Normal extensions
Galois Theory
Galois extensions and Galois groups
The Galois group as topological group
The Galois correspondence
Composita
Norm and trace
Normal bases
Solution by radicals
Determining Galois groups
The inverse Galois problem
Analogies and generalizations
Skew Fields and Central Simple Algebras
Definition and basic results
Quaternion algebras
Skew fields over R
Tensor products
Splitting fields
Reduced norms and traces
The Skolem-Noether Theorem
The Brauer group
Bibliography
Index of Notations
Index
About the Author
