Garrett Abstract Algebra

1. Auflage 2007
ISBN: 978-1-58488-690-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 472 Seiten

Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal mapping properties, rather than by constructions whose technical details are irrelevant.

Addresses Common Curricular Weaknesses
In addition to standard introductory material on the subject, such as Lagrange's and Sylow's theorems in group theory, the text provides important specific illustrations of general theory, discussing in detail finite fields, cyclotomic polynomials, and cyclotomic fields. The book also focuses on broader background, including brief but representative discussions of naive set theory and equivalents of the axiom of choice, quadratic reciprocity, Dirichlet's theorem on primes in arithmetic progressions, and some basic complex analysis. Numerous worked examples and exercises throughout facilitate a thorough understanding of the material.

Zielgruppe

Advanced undergraduate and graduate students in abstract algebra courses.

Autoren/Hrsg.

Garrett, Paul B.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

PREFACE
INTRODUCTION

THE INTEGERS
Unique factorization
Irrationalities
Z/m, the integers mod m
Fermat's little theorem
Sun-Ze's theorem
Worked examples

GROUPS I
Groups
Subgroups
Homomorphisms, kernels, normal subgroups
Cyclic groups
Quotient groups
Groups acting on sets
The Sylow theorem
Trying to classify finite groups, part I
Worked examples

THE PLAYERS: RINGS, FIELDS
Rings, fields
Ring homomorphisms
Vector spaces, modules, algebras
Polynomial rings I

COMMUTATIVE RINGS I
Divisibility and ideals
Polynomials in one variable over a field
Ideals and quotients
Ideals and quotient rings
Maximal ideals and fields
Prime ideals and integral domains
Fermat-Euler on sums of two squares
Worked examples

LINEAR ALGEBRA I: DIMENSION
Some simple results
Bases and dimension
Homomorphisms and dimension

FIELDS I
Adjoining things
Fields of fractions, fields of rational functions
Characteristics, finite fields
Algebraic field extensions
Algebraic closures

SOME IRREDUCIBLE POLYNOMIALS
Irreducibles over a finite field
Worked examples

CYCLOTOMIC POLYNOMIALS
Multiple factors in polynomials
Cyclotomic polynomials
Examples
Finite subgroups of fields
Infinitude of primes p = 1 mod n
Worked examples

FINITE FIELDS
Uniqueness
Frobenius automorphisms
Counting irreducibles

MODULES OVER PIDS
The structure theorem
Variations
Finitely generated abelian groups
Jordan canonical form
Conjugacy versus k[x]-module isomorphism
Worked examples

FINITELY GENERATED MODULES
Free modules
Finitely generated modules over a domain
PIDs are UFDs
Structure theorem, again
Recovering the earlier structure theorem
Submodules of free modules

POLYNOMIALS OVER UFDS
Gauss's lemma
Fields of fractions
Worked examples

SYMMETRIC GROUPS
Cycles, disjoint cycle decompositions
Transpositions
Worked examples

NAIVE SET THEORY
Sets
Posets, ordinals
Transfinite induction
Finiteness, infiniteness
Comparison of infinities
Example: transfinite Lagrange replacement
Equivalents of the axiom of choice

SYMMETRIC POLYNOMIALS
The theorem
First examples
A variant: discriminants

EISENSTEIN'S CRITERION
Eisenstein's irreducibility criterion
Examples

VANDERMONDE DETERMINANTS
Vandermonde determinants
Worked examples

CYCLOTOMIC POLYNOMIALS II
Cyclotomic polynomials over Z
Worked examples

ROOTS OF UNITY
Another proof of cyclicness
Roots of unity
Q with roots of unity adjoined
Solution in radicals, Lagrange resolvents
Quadratic fields, quadratic reciprocity
Worked examples

CYCLOTOMIC III
Prime-power cyclotomic polynomials over Q
Irreducibility of cyclotomic polynomials over Q
Factoring Fn(x) in Fp[x] with p n
Worked examples

PRIMES IN ARITHMETIC PROGRESSIONS
Euler's theorem and the zeta function
Dirichlet's theorem
Dual groups of abelian groups
Non-vanishing on Re(s) = 1
Analytic continuations
Dirichlet series with positive coefficients

GALOIS THEORY
Field extensions, imbeddings, automorphisms
Separable field extensions
Primitive elements
Normal field extensions
The main theorem
Conjugates, trace, norm
Basic examples
Worked examples

SOLVING EQUATIONS BY RADICALS
Galois' criterion
Composition series, Jordan-Hölder theorem
Solving cubics by radicals
Worked examples

EIGENVECTORS, SPECTRAL THEOREMS
Eigenvectors, eigenvalues
Diagonalizability, semi-simplicity
Commuting endomorphisms ST = TS
Inner product spaces
Projections without coordinates
Unitary operators
Spectral theorems
Corollaries of the spectral theorem
Worked examples

DUALS, NATURALITY, BILINEAR FORMS
Dual vector spaces
First example of naturality
Bilinear forms
Worked examples

DETERMINANTS I
Prehistory
Definitions
Uniqueness and other properties
Existence

TENSOR PRODUCTS
Desiderata
Definitions, uniqueness, existence
First examples
Tensor products f × g of maps
Extension of scalars, functoriality, naturality
Worked examples

EXTERIOR POWERS
Desiderata
Definitions, uniqueness, existence
Some elementary facts
Exterior powers ?nf of maps
Exterior powers of free modules
Determinants revisited
Minors of matrices
Uniqueness in the structure theorem
Cartan's lemma
Worked examples