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Galois Theory 4th Edition by Ian Nicholas Stewart ISBN 1482245833 9781482245837

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Galois Theory 4th Edition Ian Nicholas Stewart Digital Instant Download

Author(s): Ian Nicholas Stewart
ISBN(s): 9781482245837, 1482245833
Edition: 4
File Details: PDF, 4.11 MB
Year: 2015
Language: english
SKU: EB-11298028 Category: Tag:

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ISBN 10: 1482245833 
ISBN 13: 9781482245837
Author: Ian Nicholas Stewart

Since 1973, Galois Theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fourth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students.

New to the Fourth Edition

  • The replacement of the topological proof of the fundamental theorem of algebra with a simple and plausible result from point-set topology and estimates that will be familiar to anyone who has taken a first course in analysis
  • Revised chapter on ruler-and-compass constructions that results in a more elegant theory and simpler proofs
  • A section on constructions using an angle-trisector since it is an intriguing and direct application of the methods developed
  • A new chapter that takes a retrospective look at what Galois actually did compared to what many assume he did
  • Updated references

This bestseller continues to deliver a rigorous yet engaging treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.

Galois Theory 4th Table of contents:

Chapter 1: Classical Algebra

  • Complex Numbers
  • Subfields and Subrings of the Complex Numbers
  • Solving Equations
  • Solution by Radicals
  • Quintic Equations

Chapter 2: The Fundamental Theorem of Algebra

  • Polynomials
  • Fundamental Theorem of Algebra
  • Implications

Chapter 3: Factorisation of Polynomials

  • The Euclidean Algorithm
  • Irreducibility
  • Gauss’s Lemma
  • Eisenstein’s Criterion
  • Reduction Modulo p
  • Zeros of Polynomials

Chapter 4: Field Extensions

  • Field Extensions
  • Rational Expressions
  • Simple Extensions

Chapter 5: Simple Extensions

  • Algebraic and Transcendental Extensions
  • The Minimal Polynomial
  • Simple Algebraic Extensions
  • Classifying Simple Extensions

Chapter 6: The Degree of an Extension

  • Definition of the Degree
  • The Tower Law

Chapter 7: Ruler-and-Compass Constructions

  • Approximate Constructions and More General Instruments
  • Constructions in C
  • Specific Constructions
  • Impossibility Proofs
  • Construction From a Given Set of Points

Chapter 8: The Idea Behind Galois Theory

  • A First Look at Galois Theory
  • Galois Groups According to Galois
  • How to Use the Galois Group
  • The Abstract Setting
  • Polynomials and Extensions
  • The Galois Correspondence
  • Diet Galois
  • Natural Irrationalities

Chapter 9: Normality and Separability

  • Splitting Fields
  • Normality
  • Separability

Chapter 10: Counting Principles

  • Linear Independence of Monomorphisms

Chapter 11: Field Automorphisms

  • K-Monomorphisms
  • Normal Closures

Chapter 12: The Galois Correspondence

  • The Fundamental Theorem of Galois Theory

Chapter 13: A Worked Example

Chapter 14: Solubility and Simplicity

  • Soluble Groups
  • Simple Groups
  • Cauchy’s Theorem

Chapter 15: Solution by Radicals

  • Radical Extensions
  • An Insoluble Quintic
  • Other Methods

Chapter 16: Abstract Rings and Fields

  • Rings and Fields
  • General Properties of Rings and Fields
  • Polynomials Over General Rings
  • The Characteristic of a Field
  • Integral Domains

Chapter 17: Abstract Field Extensions

  • Minimal Polynomials
  • Simple Algebraic Extensions
  • Splitting Fields
  • Normality
  • Separability
  • Galois Theory for Abstract Fields

Chapter 18: The General Polynomial Equation

  • Transcendence Degree
  • Elementary Symmetric Polynomials
  • The General Polynomial
  • Cyclic Extensions
  • Solving Equations of Degree Four or Less

Chapter 19: Finite Fields

  • Structure of Finite Fields
  • The Multiplicative Group
  • Application to Solitaire

Chapter 20: Regular Polygons

  • What Euclid Knew
  • Which Constructions are Possible?
  • Regular Polygons
  • Fermat Numbers
  • How to Draw a Regular 17-gon

Chapter 21: Circle Division

  • Genuine Radicals
  • Fifth Roots Revisited
  • Vandermonde Revisited
  • The General Case
  • Cyclotomic Polynomials
  • Galois Group of
  • The Technical Lemma
  • More on Cyclotomic Polynomials
  • Constructions Using a Trisector

Chapter 22: Calculating Galois Groups

  • Transitive Subgroups
  • Bare Hands on the Cubic
  • The Discriminant
  • General Algorithm for the Galois Group

Chapter 23: Algebraically Closed Fields

  • Ordered Fields and Their Extensions
  • Sylow’s Theorem
  • The Algebraic Proof

Chapter 24: Transcendental Numbers

  • Irrationality
  • Transcendence of e
  • Transcendence of π

Chapter 25: What Did Galois Do or Know?

  • List of the Relevant Material
  • The First Memoir
  • What Galois Proved

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