AlgebraAlgebra 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. |
Contents
| 1 | |
| 3 | |
| 6 | |
| 7 | |
| 9 | |
| 12 | |
Cardinal Numbers | 15 |
Groups | 23 |
Modules Homomorphisms and Exact Sequences | 169 |
Free Modules and Vector Spaces | 180 |
Projective and Injective Modules | 190 |
Hom and Duality | 199 |
Tensor Products | 207 |
Modules over a Principal Ideal Domain | 218 |
Algebras | 226 |
Fields and Galois Theory | 230 |
Semigroups Monoids and Groups | 24 |
Homomorphisms and Subgroups | 30 |
Cyclic Groups | 35 |
Cosets and Counting | 37 |
Normality Quotient Groups and Homomorphisms | 41 |
Symmetric Alternating and Dihedral Groups | 46 |
Products Coproducts and Free Objects | 52 |
Direct Products and Direct Sums | 59 |
Free Groups Free Products Generators Relations | 69 |
The Structure of Groups | 72 |
Free Abelian Groups 2 Finitely Generated Abelian Groups 1 1 3 6 7 9 12 | 73 |
15 | 81 |
The KrullSchmidt Theorem | 83 |
The Action of a Group on a Set | 88 |
The Sylow Theorems | 92 |
Classification of Finite Groups | 96 |
Nilpotent and Solvable Groups | 100 |
Normal and Subnormal Series | 107 |
Rings | 114 |
Rings and Homomorphisms | 115 |
Ideals | 122 |
24 | 130 |
Factorization in Commutative Rings | 135 |
Rings of Quotients and Localization | 142 |
Rings of Polynomials and Formal Power Series | 149 |
Factorization in Polynomial Rings | 157 |
Modules | 168 |
Field Extensions | 231 |
Ruler and Compass Constructions | 238 |
The Fundamental Theorem | 243 |
Symmetric Rational Functions | 252 |
Splitting Fields Algebraic Closure and Normality | 257 |
The Fundamental Theorem of Algebra | 265 |
The Galois Group of a Polynomial | 269 |
Finite Fields | 278 |
Separability | 282 |
Cyclic Extensions | 289 |
Cyclotomic Extensions | 297 |
Radical Extensions | 302 |
The General Equation of Degree n | 307 |
The Structure of Fields | 311 |
Linear Disjointness and Separability | 318 |
Linear Algebra | 327 |
Generators and Relations | 343 |
Commutative Rings and Modules | 371 |
The Structure of Rings | 414 |
Categories | 464 |
List of Symbols | 485 |
41 | 486 |
52 | 491 |
| 492 | |
| 493 | |
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Common terms and phrases
A₁ abelian group algebraically closed ascending chain condition automorphism b₁ basis chain condition char commutative ring Consequently contains Corollary cyclic defined Definition denoted direct sum division ring divisors element endomorphism epimorphism equivalent EXAMPLE Exercise exists extension field ɛ F field F finite dimensional free module function functor G₁ Galois group given group G hence Hint implies integral domain intermediate field isomorphism K-algebra left Artinian left ideal left R-module Lemma Let F linear linearly independent matrix monic monomorphism morphism multiplicative nilpotent Noetherian nonzero normal subgroup P₁ phism positive integer prime ideal primitive principal ideal domain Proposition quotient R-module R-module homomorphism r₁ 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 u₁ vector space Verify whence zero