AlgebraAlgebra fulfills a definite need to provide a selfcontained, one volume, graduate level algebra text that is readable by the average graduate student and flexible enough to accomodate a wide variety of instructors and course contents. The guiding philosophical principle throughout the text is that the material should be presented in the maximum usable generality consistent with good pedagogy. Therefore it is essentially selfcontained, stresses clarity rather than brevity and contains an unusually large number of illustrative exercises. The book covers major areas of modern algebra, which is a necessity for most mathematics students in sufficient breadth and depth. 
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Review: Algebra (Graduate Texts in Mathematics) (Graduate Texts in Mathematics #73)
User Review  Han Zhicheng  GoodreadsThe firstever GTM text in my life. Hopefully this will kick a good start. This book is excellent for those beginner in algebra who would like to have a panorama of this field. Its settings are clear ... Read full review
Review: Algebra (Graduate Texts in Mathematics) (Graduate Texts in Mathematics #73)
User Review  GoodreadsThe firstever GTM text in my life. Hopefully this will kick a good start. This book is excellent for those beginner in algebra who would like to have a panorama of this field. Its settings are clear ... Read full review
Contents
Groups  23 
1 Semigroups Monoids and Groups  24 
2 Homomorphisms and Subgroups  30 
3 Cyclic Groups  35 
4 Cosets and Counting  37 
5 Normality Quotient Groups and Homomorphisms  41 
6 Symmetric Alternating and Dihedral Groups  46 
Products Coproducts and Free Objects  52 
4 The Galois Group of a Polynomial  269 
5 Finite Fields  278 
6 Separability  282 
7 Cyclic Extensions  289 
8 Cyclotomic Extensions  297 
9 Radical Extensions  302 
The Structure of Fields  311 
2 Linear Disjointness and Separability  318 
8 Direct Products and Direct Sums  59 
9 Free Groups Free Products Generators Relations  64 
The Structure of Groups  70 
2 Finitely Generated Abelian Groups  76 
3 The KrullSchmidt Theorem  83 
4 The Action of a Group on a Set  88 
5 The Sylow Theorems  92 
6 Classification of Finite Groups  96 
7 Nilpotent and Solvable Groups  100 
8 Normal and Subnormal Series  107 
Rings  114 
1 Rings and Homomorphisms  115 
2 Ideals  122 
3 Factorization in Commutative Rings  135 
4 Rings of Quotients and Localization  142 
5 Ring of Polynomials and Formal Power Series  149 
6 Factorization in Polynomial Rings  157 
Modules  168 
1 Modules Homomorphisms and Exact Sequences  169 
2 Free Modules and Vector Spaces  180 
3 Projective and Injective Modules  190 
4 Hom and Duality  199 
5 Tensor Products  207 
6 Modules Over a Principal Ideal Domain  218 
7 Algebras  226 
Fields and Galois Theory  230 
1 Field Extensions  231 
2 The Fundamental Theorem  243 
3 Splitting Fields Algebraic Closure and Normality  257 
Linear Algebra  327 
1 Matrices and Maps  328 
2 Rank and Equivalence  335 
3 Determinants  348 
4 Decomposition of a Single Linear Transformation and Similarity  355 
5 The Characteristic Polynomial Eigenvectors and Eigenvalues  366 
Commutative Rings and Modules  371 
1 Chain Conditions  372 
2 Prime and Primary Ideals  377 
3 Primary Decomposition  383 
4 Noetherian Rings and Modules  387 
5 Ring Extensions  394 
6 Dedekind Domains  400 
7 The Hilbert Nullstellensatz  409 
The Structure of Rings  414 
1 Simple and Primitive Rings  415 
2 The Jacobson Radical  424 
3 Semisimple Rings  434 
4 The Prime Radical Prime and Semiprime Rings  444 
5 Algebras  450 
6 Division Algebras  456 
Categories  464 
1 Functors and Natural Transformations  465 
2 Adjoint Functors  476 
3 Morphisms  480 
List of Symbols  485 
489  
493  
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Common terms and phrases
abelian group algebraically closed ascending chain condition automorphism basis bijection chain condition char commutative ring Consequently contains Corollary coset cyclic defined Definition denoted direct product direct sum disjoint division ring divisors endomorphism epimorphism equivalent EXAMPLE Exercise exists extension field finite dimensional finite number free module function functor Galois group given group G hence Hint implies infinite integral domain intermediate field invertible isomorphism left Artinian left ideal left module Lemma Let F Let G linear linearly independent matrix module homomorphism monic monomorphism morphism multiplicative nilpotent Noetherian nonempty normal subgroup phism positive integer prime ideal primitive principal ideal domain proof of Theorem Proposition prove quotient Rmodule radical resp ring with identity Section semisimple SKETCH OF PROOF solvable splitting field subfield subgroup of G submodule subring subset Sylow Theorem 1.6 transcendence base vector space Verify whence zero