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Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics)
Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) Summary:By Mark Steinberger
The intent of this book is to introduce readers to algebra from a point of view that stresses examples and classification. Whenever possible, the main theorems are treated as tools that may be used to construct and analyze specific types of groups, rings, fields, modules, etc. Sample constructions and classifications are given in both text and exercises. The first chapter gives a summary of the basic set theory that is used throughout the text.Chapter 2 introduces groups and homomorphisms, and gives some examples that will be used as illustrations throughout the material on group theory. Chapter 3 develops symmetric groups and the theory of G-sets, giving some useful counting arguments for classifying low order groups. Actions of one group on another via automorphisms, with conjugation as the initial example, are emphasized. Chapter 4 studies consequences of normality, beginning with the Noether Isomorphism Theorems and following with a study of simple groups and composition series, and the classification of all finite abelian groups via the Fundamental Theorem of Finite Abelian Groups. The automorphism group of a cyclic group is calculated. Then, semidirect products are introduced, using the calculations of automorphism groups to construct numerous examples that will be essential in classifying low order groups. The chapter closes with a study of extensions of groups, developing some useful classification tools. Chapter 5 develops some additional tools, such as the Sylow Theorems, and applies the theory to the classification of groups of many of the orders ≤ 63, in both text and exercises. Solvable and nilpotent groups are also studied. Chapter 6 is an introduction to basic category theoretic notions, with examples drawn from the earlier material. The concepts developed here are useful in understanding rings and modules, though they are used sparingly in the text on that material. Pushouts of groups are constructed and the technique of generators and relations is given. Chapter 7 provides a general introduction to the theory of rings and modules. Examples are introduced, including the quaternions, the p-adic integers, the cyclotomic integers and rational numbers, polynomials, group rings, and more. Free modules and chain conditions are studied, and the elementary theory of vector spaces and matrices is developed. The chapter closes with the study of rings and modules of fractions, which will be applied to the study of P.I.D.s and fields. Chapter 8 develops the theory of P.I.D.s and covers applications to field theory. The exercises treat prime factorization in some particular Euclidean number rings. The basic theory of algebraic and transcendental field extensions is given, including many of the basic tools to be used in Galois theory. The cyclotomic polynomials over Q are calculated, and the chapter closes with a presentation of the Fundamental Theorem of Finitely Generated Modules over a P.I.D. Chapter 9 gives some of the basic tools of ring and module theory: Nakayama’s Lemma, primary decomposition, tensor products, extension of rings, projectives, and the exactness properties of tensor and hom. In Section 9.3, Hilbert’s Nullstellensatz is proven, using many of the tools so far developed, and is used to define algebraic varieties. In Section 9.5, extension of rings is used to extend some of the standard results about injections and surjections between finite dimensional vector spaces to the study of maps between finitely generated free modules over more general rings. Also, algebraic K-theory is introduced with the study of K₀. Chapter 10 gives a formal development of linear algebra, culminating in the classification of matrices via canonical forms. This is followed by some more general material about general linear groups, followed by an introduction to K₁. Chapter 11 is devoted to Galois theory. The usual applications are given, e.g. the Primitive Element Theorem, the Fundamental Theorem of Algebra, and the classification of finite fields. Sample calculations of Galois groups are given in text and exercises, particularly for splitting fields of polynomials of the form Xⁿ - a The insolvability of polynomials of degree ≥ 5 is treated. Chapter 12 gives the theory of hereditary and semisimple rings with an emphasis on Dedekind domains and group algebras. Stable and unstable classifications of projective modules are given for Dedekind domains and semisimple rings. Contents: Chapter 1. A Little Set Theory 1.1. Properties of Functions 1.2. Factorizations of Functions 1.3. Relations 1.4. Equivalence Relations 1.5. Generating an Equivalence Relation 1.6. Cartesian Products 1.7. Formalities about Functions Chapter 2. Groups: Basic Definitions and Examples 2.1. Groups and Monoids 2.2. Subgroups 2.3. The Subgroups of the Integers 2.4. Finite Cyclic Groups: Modular Arithmetic 2.5. Homomorphisms and Isomorphisms 2.6. The Classification Problem 2.7. The Group of Rotations of the Plane 2.8. The Dihedral Groups 2.9. Quaternions 2.10. Direct Products Chapter 3. G-sets and Counting 3.1. Symmetric Groups: Cayley's Theorem 3.2. Cosets and Index: Lagrange's Theorem 3.3. G-sets and Orbits 3.4. Supports of Permutations 3.5. Cycle Structure 3.6. Conjugation and Other Automorphisms 3.7. Conjugating Subgroups: Normality Chapter 4. Normality and Factor Groups 4.1. The Noether Isomorphism Theorems 4.2. Simple Groups 4.3. The Jordan-Hölder Theorem 4.4. Abelian Groups: the Fundamental Theorem 4.5. The Automorphisms of a Cyclic Group 4.6. Semidirect Products 4.7. Extensions Chapter 5. Sylow Theory, Solvability, and Classification 5.1. Cauchy's Theorem 5.2. p-Groups 5.3. Sylow Subgroups 5.4. Commutator Subgroups and Abelianization 5.5. Solvable Groups 5.6. Hall's Theorem 5.7. Nilpotent Groups 5.8. Matrix Groups Chapter 6. Categories in Group Theory 6.1. Categories 6.2. Functors 6.3. Universal Mapping Properties: Products and Coproducts 6.4. Pushouts and Pullbacks 6.5. Infinite Products and Coproducts 6.6. Free Functors 6.7. Generators and Relations 6.8. Direct and Inverse Limits 6.9. Natural Transformations and Adjoints 6.10. General Limits and Colimits Chapter 7. Rings and Modules 7.1. Rings 7.2. Ideals 7.3. Polynomials 7.4. Symmetry of Polynomials 7.5. Group Rings and Monoid Rings 7.6. Ideals in Commutative Rings 7.7. Modules 7.8. Chain Conditions 7.9. Vector Spaces 7.10. Matrices and Transformations 7.11. Rings of Fractions Chapter 8. P.I.D.s and Field Extensions 8.1. Euclidean Rings, P.I.D.s, and U.F.D.s 8.2. Algebraic Extensions 8.3. Transcendence Degree 8.4. Algebraic Closures 8.5. Criteria for Irreducibility 8.6. The Frobenius map 8.7. Repeated Roots 8.8. Cyclotomic Polynomials 8.9. Modules over P.I.D.s Chapter 9. Radicals, Tensor Products, and Exactness 9.1. Radicals 9.2. Primary Decomposition 9.3. The Nullstellensatz and the Prime Spectrum 9.4. Tensor Products 9.5. Tensor Products and Exactness 9.6. Tensor Products of Algebras 9.7. The Hom Functors 9.8. Projective Modules 9.9. The Grothendieck Construction: K₀ 9.10. Tensor Algebras and Their Relatives Chapter 10. Linear Algebra 10.1. Traces 10.2. Multilinear Alternating Forms 10.3. Properties of Determinants 10.4. The Characteristic Polynomial 10.5. Eigenvalues and Eigenvectors 10.6. The Classification of Matrices 10.7. Jordan Canonical Form 10.8. Generators for Matrix Groups 10.9. K₁ Chapter 11. Galois Theory 11.1. Embeddings of Fields 11.2. Normal Extensions 11.3. Finite Fields 11.4. Separable Extensions 11.5. Galois Theory 11.6. The Fundamental Theorem of Algebra 11.7. Cyclotomic Extensions 11.8. n-th Roots 11.9. Cyclic Extensions 11.10. Kummer Theory 11.11. Solvable Extensions 11.12. The General Equation 11.13. Normal Bases 11.14. Norms and Traces Chapter 12. Hereditary and Semisimple Rings 12.1. Maschke's Theorem and Projectives 12.2. Semisimple Rings 12.3. Jacobson Semisimplicity 12.4. Homological Dimension 12.5. Hereditary Rings 12.6. Dedekind Domains 12.7. Integral Dependence Bibliography password:baribal Please select one mirror to download
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Sponsored LinksAlgebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) Keywordsclassification polynomials tensor finite fields examples algebra tools subgroups galois algebraic fundamental p i d s relations semisimple exercises matrices cyclic cyclotomic chapter develops chapter closes finite abelian abelian groups prindle weber finitely generated fundamental theorem vector spaces fields chapter exercises treat finite abelian groupsAlgebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) download copyrightThis site does not store Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) on its server. We only index and link to Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) provided by other sites. Please contact the content providers to delete Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) if any and email us, we'll remove relevant links or contents immediately. |
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