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Publisher:

Dover Publications

Publication Date:

2009

Number of Pages:

686

Format:

Paperback

Edition:

2

Price:

25.95

ISBN:

9780486471877

Category:

Textbook

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Allen Stenger

07/5/2011

This book is a sequel to the author’s Basic Algebra I, but it is not a continuation: it treats a completely new set of topics, at a much higher level of abstraction. It is “basic” only in the sense that the material studied here is the basis for getting started with modern abstract algebra. The present volume is an unaltered reprint of the 1989 second edition.

The first chapter covers categories, and most of the rest of the book takes a category-theoretic approach. The pace here is more leisurely than it was in the first volume, and the book does not go into great depth on any topic. For example, the chapter on valuations does not go much beyond the ramification index, Krasner’s Lemma, and Hensel’s Lemma. The book does have a large number of exercises, of varying levels of difficulty, and most of these explore further developments and are not simple applications of the text material.

The great weakness of the book is that it is relentlessly abstract: there are almost no concrete examples of the abstract structures that are being studied. Each chapter is careful to credit the areas of mathematics from which it is abstracted (usually these are number theory and algebraic geometry), but we never see any applications back to the source subjects.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

- Preface
- Preface to the First Edition
- INTRODUCTION
- 0.1 Zorn's lemma
- 0.2 Arithmetic of cardinal numbers
- 0.3 Ordinal and cardinal numbers
- 0.4 Sets and classes
- References

- CATEGORIES
- 1.1 Definition and examples of categories
- 1.2 Some basic categorical concepts
- 1.3 Functors and natural transformations
- 1.4 Equivalence of categories
- 1.5 Products and coproducts
- 1.6 The hom functors. Representable functors
- 1.7 Universals
- 1.8 Adjoints
- References

- UNIVERSAL ALGEBRA
- 2.1 Ω-algebras
- 2.2 Subalgebras and products
- 2.3 Homomorphisms and congruences
- 2.4 The lattice of congruences. Subdirect products
- 2.5 Direct and inverse limits
- 2.6 Ultraproducts
- 2.7 Free Ω-algebras
- 2.8 Varieties
- 2.9 Free products of groups
- 2.10 Internal characterization of varieties
- References

- MODULES
- 3.1 The categories
**R-mod**and**mod-R** - 3.2 Artinian and Noetherian modules
- 3.3 Schreier refinement theorem. Jordan-Hölder theorem
- 3.4 The Krull-Schmidt theorem
- 3.5 Completely reducible modules
- 3.6 Abstract dependence relations. Invariance of dimensionality
- 3.7 Tensor products of modules
- 3.8 Bimodules
- 3.9 Algebras and coalgebras
- 3.10 Projective modules
- 3.11 Injective modules. Injective hull
- 3.12 Morita contexts
- 3.13 The Wedderburn-Artin theorem for simple rings
- 3.14 Generators and progenerators
- 3.15 Equivalence of categories of modules
- References

- 3.1 The categories
- BASIC STRUCTURE THEORY OF RINGS
- 4.1 Primitivity and semi-primitivity
- 4.2 The radical of a ring
- 4.3 Density theorems
- 4.4 Artinian rings
- 4.5 Structure theory of algebras
- 4.6 Finite dimensional central simple algebras
- 4.7 The Brauer group
- 4.8 Clifford algebras
- References

- CLASSICAL REPRESENTATION THEORY OF FINITE GROUPS
- 5.1 Representations and matrix representations of groups
- 5.2 Complete reducibility
- 5.3 Application of the representation theory of algebras
- 5.4 Irreducible representations of
*S*_{n} - 5.5 Characters. Orthogonality relations
- 5.6 Direct products of groups. Characters of abelian groups
- 5.7 Some arithmetical considerations
- 5.8 Burnside's
*p*theorem^{a}q^{b} - 5.9 Induced modules
- 5.10 Properties of induction. Frobenius reciprocity theorem
- 5.11 Further results on induced modules
- 5.12 Brauer's theorem on induced characters
- 5.13 Brauer's theorem on splitting fields
- 5.14 The Schur index
- 5.15 Frobenius groups
- References

- ELEMENTS OF HOMOLOGICAL ALGEBRA WITH APPLICATIONS
- 6.1 Additive and abelian categories
- 6.2 Complexes and homology
- 6.3 Long exact homology sequence
- 6.4 Homotopy
- 6.5 Resolutions
- 6.6 Derived functors
- 6.7 Ext
- 6.8 Tor
- 6.9 Cohomology of groups
- 6.10 Extensions of groups
- 6.11 Cohomology of algebras
- 6.12 Homological dimension
- 6.13 Koszul's complex and Hilbert's syzygy theorem
- References

- COMMUTATIVE IDEAL THEORY: GENERAL THEORY AND NOETHERIAN RINGS
- 7.1 Prime ideals. Nil radical
- 7.2 Localization of rings
- 7.3 Localization of modules
- 7.4 Localization at the complement of a prime ideal. Local-global relations
- 7.5 Prime spectrum of a commutative ring
- 7.6 Integral dependence
- 7.7 Integrally closed domains
- 7.8 Rank of projective modules
- 7.9 Projective class group
- 7.10 Noetherian rings
- 7.11 Commutative artinian rings
- 7.12 Affine algebraic varieties. The Hilbert Nullstellensatz
- 7.13 Primary decompositions
- 7.14 Artin-Rees lemma. Krull intersection theorem
- 7.15 Hilbert's polynomial for a graded module
- 7.16 The characteristic polynomial of a noetherian local ring
- 7.17 Krull dimension
- 7.18
*I*-adic topologies and completions - References

- FIELD THEORY
- 8.1 Algebraic closure of a field
- 8.2 The Jacobson-Bourbaki correspondence
- 8.3 Finite Galois theory
- 8.4 Crossed products and the Brauer group
- 8.5 Cyclic algebras
- 8.6 Infinite Galois theory
- 8.7 Separability and normality
- 8.8 Separable splitting fields
- 8.9 Kummer extensions
- 8.10 Rings of Witt vectors
- 8.11 Abelian
*p*-extension - 8.12 Transcendency bases
- 8.13 Transcendency bases for domains. Affine algebras
- 8.14 Luroth's theorem
- 8.15 Separability for arbitrary extension fields
- 8.16 Derivations
- 8.17 Galois theory for purely inseparable extensions of exponent one
- 8.18 Tensor products of fields
- 8.19 Free composites of fields
- References

- VALUATION THEORY
- 9.1 Absolute values
- 9.2 The approximation theorem
- 9.3 Absolute values on
*Q*and*F*(*x*) - 9.4 Completion of a field
- 9.5 Finite dimensional extensions of complete fields. The archimedean case
- 9.6 Valuations
- 9.7 Valuation rings and places
- 9.8 Extension of homomorphisms and valuations
- 9.9 Determination of the absolute values of a finite dimensional extension field
- 9.10 Ramification index and residue degree. Discrete valuations
- 9.11 Hensel's lemma
- 9.12 Local fields
- 9.13 Totally disconnected locally compact division rings
- 9.14 The Brauer group of a local field
- 9.15 Quadratic forms over local fields
- References

- DEDEKIND DOMAINS
- 10.1 Fractional ideals. Dedekind domains
- 10.2 Characterizations of Dedekind domains
- 10.3 Integral extensions of Dedekind domains
- 10.4 Connections with valuation theory
- 10.5 Ramified primes and the discriminant
- 10.6 Finitely generated modules over a Dedekind domain
- References

- FORMALLY REAL FIELDS
- 11.1 Formally real fields
- 11.2 Real closures
- 11.3 Totally positive elements
- 11.4 Hilbert's seventeenth problem
- 11.5 Pfister theory of quadratic forms
- 11.6 Sums of squares in
*R*(*x*, ...,_{1}*x*),_{n}*R*a real closed field - 11.7 Artin-Schreier characterization of real closed fields
- References

- INDEX

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