Abstract Algebra: 1st Edition (Hardback) book cover

Abstract Algebra

1st Edition

By Paul B. Garrett

Chapman and Hall/CRC

472 pages | 11 B/W Illus.

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Hardback: 9781584886891
pub: 2007-09-25
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Description

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.

Table of Contents

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

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General