Basic Algebra 1st Edition by Anthony W. Knapp – Ebook PDF Instant Download/Delivery: 0817632484, 9780817632489
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ISBN 10: 0817632484
ISBN 13: 9780817632489
Author: Anthony W. Knapp
Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them. The presentation includes blocks of problems that introduce additional topics and applications to science and engineering to guide further study. Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems. Basic Algebra presents the subject matter in a forward-looking way that takes into account its historical development. It is suitable as a text in a two-semester advanced undergraduate or first-year graduate sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs.
Basic Algebra 1st Table of contents:
CHAPTER I Preliminaries about the Integers, Polynomials, and Matrices
1. Division and Euclidean Algorithms
2. Unique Factorization of Integers
3. Unique Factorization of Polynomials
4. Permutations and Their Signs
5. Row Reduction
6. Matrix Operations
7. Problems
CHAPTER II Vector Spaces over Q, R, and C
1. Spanning, Linear Independence, and Bases
2. Vector Spaces Defined by Matrices
3. Linear Maps
4. Dual Spaces
5. Quotients of Vector Spaces
6. Direct Sums and Direct Products of Vector Spaces
7. Determinants
8. Eigenvectors and Characteristic Polynomials
9. Bases in the Infinite-Dimensional Case
10. Problems
CHAPTER III Inner-Product Spaces
1. Inner Products and Orthonormal Sets
2. Adjoints
3. Spectral Theorem
4. Problems
CHAPTER IV Groups and Group Actions
1. Groups and Subgroups
2. Quotient Spaces and Homomorphisms
3. Direct Products and Direct Sums
4. Rings and Fields
5. Polynomials and Vector Spaces
6. Group Actions and Examples
7. Semidirect Products
8. Simple Groups and Composition Series
9. Structure of Finitely Generated Abelian Groups
10. Sylow Theorems
11. Categories and Functors
12. Problems
CHAPTER V Theory of a Single Linear Transformation
1. Introduction
2. Determinants over Commutative Rings with Identity
3. Characteristic and Minimal Polynomials
4. Projection Operators
5. Primary Decomposition
6. Jordan Canonical Form
7. Computations with Jordan Form
8. Problems
CHAPTER VI Multilinear Algebra
1. Bilinear Forms and Matrices
2. Symmetric Bilinear Forms
3. Alternating Bilinear Forms
4. Hermitian Forms
5. Groups Leaving a Bilinear Form Invariant
6. Tensor Product of Two Vector Spaces
7. Tensor Algebra
8. Symmetric Algebra
9. Exterior Algebra
10. Problems
CHAPTER VII Advanced Group Theory
1. Free Groups
2. Subgroups of Free Groups
3. Free Products
4. Group Representations
5. Burnside’s Theorem
6. Extensions of Groups
7. Problems
CHAPTER VIII Commutative Rings and Their Modules
1. Examples of Rings and Modules
2. Integral Domains and Fields of Fractions
3. Prime and Maximal Ideals
4. Unique Factorization
5. Gauss’s Lemma
6. Finitely Generated Modules
7. Orientation for Algebraic Number Theory and Algebraic Geometry
8. Noetherian Rings and the Hilbert Basis Theorem
9. Integral Closure
10. Localization and Local Rings
11. Dedekind Domains
12. Problems
CHAPTER IX Fields and Galois Theory
1. Algebraic Elements
2. Construction of Field Extensions
3. Finite Fields
4. Algebraic Closure
5. Geometric Constructions by Straightedge and Compass
6. Separable Extensions
7. Normal Extensions
8. Fundamental Theorem of Galois Theory
9. Application to Constructibility of Regular Polygons
10. Application to Proving the Fundamental Theorem of Algebra
11. Application to Unsolvability of Polynomial Equations with Nonsolvable Galois Group
12. Construction of Regular Polygons
13. Solution of Certain Polynomial Equations with Solvable Galois Group
14. Proof That π Is Transcendental
15. Norm and Trace
16. Splitting of Prime Ideals in Extensions
17. Two Tools for Computing Galois Groups
18. Problems
CHAPTER X Modules over Noncommutative Rings
1. Simple and Semisimple Modules
2. Composition Series
3. Chain Conditions
4. Hom and End for Modules
5. Tensor Product for Modules
6. Exact Sequences
7. Problems
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