Galois theory. (2015)

• Record Type: Book
• Further Information: Note: I.N. Stewart.
• Authors: Stewart, Ian, 1945-
• Contents: Classical Algebra ; Complex Numbers; Subfields and Subrings of the Complex Numbers; Solving Equations; Solution by Radicals The Fundamental Theorem of Algebra; Polynomials; Fundamental Theorem of Algebra; Implications Factorisation of Polynomials; The Euclidean Algorithm; Irreducibility; Gauss’s Lemma; Eisenstein’s Criterion; Reduction Modulo p ; Zeros of Polynomials Field Extensions; Field Extensions; Rational Expressions; Simple Extensions Simple Extensions; Algebraic and Transcendental Extensions; The Minimal Polynomial; Simple Algebraic Extensions; Classifying Simple Extensions The Degree of an Extension; Definition of the Degree; The Tower Law Ruler-and-Compass Constructions; Approximate Constructions and More General Instruments; Constructions in C; Specific Constructions; Impossibility Proofs ; Construction from a Given Set of Points 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 Normality and Separability; Splitting Fields; Normality; Separability Counting Principles; Linear Independence of Monomorphisms Field Automorphisms; K -Monomorphisms; Normal Closures The Galois Correspondence; The Fundamental Theorem of Galois Theory A Worked Example Solubility and Simplicity; Soluble Groups; Simple Groups; Cauchy’s Theorem Solution by Radicals; Radical Extensions; An Insoluble Quintic; Other Methods Abstract Rings and Fields; Rings and Fields; General Properties of Rings and Fields; Polynomials over General Rings; The Characteristic of a Field; Integral Domains Abstract Field Extensions; Minimal Polynomials; Simple Algebraic Extensions . ; Splitting Fields; Normality; Separability; Galois Theory for Abstract Fields The General Polynomial Equation; Transcendence Degree; Elementary Symmetric Polynomials; The General Polynomial; Cyclic Extensions; Solving Equations of Degree Four or Less Finite Fields; Structure of Finite Fields; The Multiplicative Group; Application to Solitaire Regular Polygons; What Euclid Knew; Which Constructions Are Possible?; Regular Polygons; Fermat Numbers; How to Draw a Regular 17-gon Circle Division; Genuine Radicals; Fifth Roots Revisited; Vandermonde Revisited; The General Case; Cyclotomic Polynomials; Galois Group of Q(ζ) : Q; The Technical Lemma; More on Cyclotomic Polynomials; Constructions Using a Trisector Calculating Galois Groups; Transitive Subgroups; Bare Hands on the Cubic; The Discriminant; General Algorithm for the Galois Group Algebraically Closed Fields; Ordered Fields and Their Extensions; Sylow’s Theorem; The Algebraic Proof Transcendental Numbers; Irrationality; Transcendence of e; Transcendence of pi What Did Galois Do or Know?; List of the Relevant Material; The First Memoir; What Galois Proved; What Is Galois up to?; Alternating Groups, Especially A5 ; Simple Groups Known to Galois; Speculations about Proofs References; Index
• Edition: 4th edition
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2015
• Extent: 1 online resource, illustrations (black and white)
• Subjects: 512.32; Galois theory
• Languages: English
• ISBNs: 9781482245851; 9781482245837; 9781482245844
• Related ISBNs: 9781482245820
• Notes: Note: Includes bibliographical references and index.; Note: Description based on CIP data; item not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.