Algebra Abstract and Modern
Algebra: Abstract and Modern, spread across 16 chapters, introduces the reader to the preliminaries of algebra and then explains topics like group theory and field theory in depth. It also features a blend of numerous challenging exercises and examples that further enhance each chapter. Covering all...
| Autor principal: | |
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| Otros Autores: | |
| Formato: | Libro electrónico |
| Idioma: | Inglés |
| Publicado: |
Noida :
Pearson India
2012.
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| Materias: | |
| Ver en Biblioteca Universitat Ramon Llull: | https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009815724506719 |
Tabla de Contenidos:
- Cover
- Contents
- Preface
- Part I: Preliminaries
- Chapter 1: Sets and Relations
- 1.1 Sets and subsets
- 1.2 Relations and functions
- 1.3 Equivalence relations and partitions
- 1.4 The cardinality of a set
- Chapter 2: Number Systems
- 2.1 Integers
- 2.2 Congruence modulo n
- 2.3 Rational, real and complex numbers
- 2.4 Ordering
- 2.5 Matrices
- 2.6 Determinants
- Part II: Group Theory
- Chapter 3: Groups
- 3.1 Binary systems
- 3.2 Groups
- 3.3 Elementary properties of groups
- 3.4 Finite groups and group tables
- Chapter 4: Subgroups and Quotient Groups
- 4.1 Subgroups
- 4.2 Cyclic groups
- 4.3 Cosets of a subgroup
- 4.4 Lagrange's theorem
- 4.5 Normal subgroups
- 4.6 Quotient groups
- Chapter 5: Homomorphisms of Groups
- 5.1 Definition and examples
- 5.2 Fundamental theorem of homomorphisms
- 5.3 Isomorphism theorems
- 5.4 Automorphisms
- Chapter 6: Permutation Groups
- 6.1 Cayley's theorem
- 6.2 The symmetric group Sn
- 6.3 Cycles
- 6.4 Alternating group An and dihedral group Dn
- Chapter 7: Group Actions on Sets
- 7.1 Action of a group on a set
- 7.2 Orbits and stabilizers
- 7.3 Certain counting techniques
- 7.4 Cauchy and Sylow theorems
- Chapter 8: Structure Theory of Groups
- 8.1 Direct products
- 8.2 Finitely generated abelian groups
- 8.3 Invariants of finite abelian groups
- 8.4 Groups of small order
- Part III: Ring Theory
- Chapter 9: Rings
- 9.1 Examples and elementary properties
- 9.2 Certain special elements in rings
- 9.3 The characteristic of a ring
- 9.4 Subrings
- 9.5 Homomorphisms of rings
- 9.6 Certain special types of rings
- 9.7 Integral domains and fields
- Chapter 10: Ideals and Quotient Rings
- 10.1 Ideals
- 10.2 Quotient rings
- 10.3 Chinese remainder theorem
- 10.4 Prime ideals
- 10.5 Maximal ideals
- 10.6 Embeddings of rings.
- Chapter 11: Polynomial Rings
- 11.1 Rings of polynomials
- 11.2 The division algorithm
- 11.3 Polynomials over a field
- 11.4 Irreducible polynomials
- Chapter 12: Factorization in Integral Domains
- 12.1 Divisibility in integral domains
- 12.2 Principal ideal domains
- 12.3 Unique factorization domains
- 12.4 Polynomials over UFDs
- 12.5 Euclidean domains
- 12.6 Some applications to number theory
- Chapter 13: Modules and Vector Spaces
- 13.1 Modules and submodules
- 13.2 Homomorphisms and quotients of modules
- 13.3 Direct products and sums
- 13.4 Simple and completely reducible modules
- 13.5 Free modules
- 13.6 Vector spaces
- Part IV: Field Theory
- Chapter 14: Extension Fields
- 14.1 Extensions of a field
- 14.2 Algebraic extensions
- 14.3 Algebraically closed fields
- 14.4 Derivatives and multiple roots
- 14.5 Finite fields
- Chapter 15: Galois Theory
- 15.1 Separable and normal extensions
- 15.2 Automorphism groups and fixed fields
- 15.3 Fundamental theorem of Galois theory
- Chapter 16: Selected Applications of Galois Theory
- 16.1 Fundamental theorem of algebra
- 16.2 Cyclic extensions
- 16.3 Solvable groups
- 16.4 Polynomials solvable by radicals
- 16.5 Constructions by ruler and compass
- Answers/Hints to Selected Even-Numbered Exercises
- Index.
