Exploratory Galois theory /

Combining a concrete perspective with an exploration-based approach, Exploratory Galois Theory develops Galois theory at an entirely undergraduate level. The text grounds the presentation in the concept of algebraic numbers with complex approximations and assumes of its readers only a first course i...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Swallow, John, 1970-
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge, UK ; New York : Cambridge University Press, 2004.
Temas:
Galois theory.
Théorie de Galois.
MATHEMATICS > Algebra > Intermediate.
Galois theory
Galois-Theorie
Álgebra.
Teoria de galois.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Half-title
  • Title
  • Copyright
  • Dedication
  • Contents
  • Preface
  • Introduction
  • CHAPTER ONE Preliminaries
  • 1. Polynomials, Polynomial Rings, Factorization, and Roots in C
  • 2. Computation with Roots and Factorizations: Maple and Mathematica
  • 2.1. Approximating Roots
  • 2.2. Factoring Polynomials over Q
  • 2.3. Executing the Division Algorithm over Q
  • 2.4. Executing the Euclidean Algorithm over Q
  • 3. Ring Homomorphisms, Fields, Monomorphisms, and Automorphisms
  • 4. Groups, Permutations, and Permutation Actions
  • 5. Exercises.
  • CHAPTER TWO Algebraic Numbers, Field Extensions, and Minimal Polynomials
  • 6. The Property of Being Algebraic
  • 7. Minimal Polynomials
  • 8. The Field Generated by an Algebraic Number
  • 8.1. Rings and Vector Spaces Associated to an Algebraic Number
  • 8.2. The Ring Is a Field
  • 8.3. These Fields Are Isomorphic to Quotients of Polynomial Rings
  • 9. Reduced Forms in Q(Ü): Maple and Mathematica
  • 10. Exercises
  • CHAPTER THREE Working with Algebraic Numbers, Field Extensions, and Minimal Polynomials
  • 11. Minimal Polynomials Are Associated to Which Algebraic Numbers?
  • 11.1. A Polynomial of Degree n Has at Most n Roots in Any Field Extension
  • 11.2. A Polynomial of Degree n Factors into n Linear Factors over C
  • 11.3. Minimal Polynomials Are Minimal Polynomials for n Distinct Algebraic Numbers
  • 12. Which Algebraic Numbers Generate a Generated Field?
  • 12.1. Degrees of Minimal Polynomials of Algebraic Numbers Generating a Given Field
  • 12.2. If an Algebraic Number Generates a Field, So Do Its Affine Translations
  • 12.3. Degrees of Minimal Polynomials Divide the Dimension of an Enclosing Field.
  • 12.4. The Set of Algebraic Numbers Is Closed Under Field Operations
  • 13. Exercise Set 1
  • 14. Computation in Algebraic Number Fields: Maple and Mathematica
  • 14.1. Declaring a Field
  • 14.2. Reduced Forms
  • 14.3. Factoring Polynomials over a Field
  • 14.4. The Division Algorithm and Reduced Forms
  • 14.5. The Euclidean Algorithm and Inverses
  • 14.6. Representing Algebraic Numbers and Finding Minimal Polynomials and Factors
  • 14.7. Reduced Forms over Subfields
  • 15. Exercise Set 2
  • CHAPTER FOUR Multiply Generated Fields
  • 16. Fields Generated by Several Algebraic Numbers.
  • 16.1. Generation by Two Algebraic Numbers Is Generation by One
  • 16.2. From Multiply Generated Extensions to Multivariate Polynomial Rings
  • 16.3. Fields Generated by a Finite Number of Algebraic Numbers Are Quotients of Polynomial Rings
  • 16.4. Splitting Fields
  • 17. Characterizing Isomorphisms between Fields: Three Cubic Examples
  • 18. Isomorphisms from Multiply Generated Fields
  • 18.1. Conditions for Isomorphisms from Multiply Generated Fields
  • 18.2. Isomorphisms of Splitting Fields over Isomorphic Fields.

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