# Algebra

Springer Science & Business Media, 1974 - Mathematics - 502 pages
Algebra fulfills a definite need to provide a self-contained, 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 self-contained, 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)

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### 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 Bibliography 489 Index 493 Copyright

### Popular passages

Page 1 - P and Q” is true if both P and Q are true and false otherwise.
Page 13 - Zorn's Lemma. If A is a nonempty partially ordered set such that every chain in A has an upper bound in A, then A contains a maximal element.

Thomas W. Hungerford received his M.S. and Ph.D. from the University of Chicago. He has taught at the University of Washington and at Cleveland State University, and is now at St. Louis University. His research fields are algebra and mathematics education. He is the author of many notable books for undergraduate and graduate level courses. In addition to ABSTRACT ALGEBRA: AN INTRODUCTION, these include: ALGEBRA (Springer, Graduate Texts in Mathematics, #73. 1974); MATHEMATICS WITH APPLICATIONS, Tenth Edition (Pearson, 2011; with M. Lial and J. Holcomb); and CONTEMPORARY PRECALCULUS, Fifth Edition (Cengage, 2009; with D. Shaw).