Galois Theory

Classical Algebra Complex Numbers Subfields and Subrings of the Complex Numbers Solving Equations Solution by RadicalsThe Fundamental Theorem of Algebra Polynomials Fundamental Theorem of Algebra ImplicationsFactorisation of Polynomials The Euclidean Algorithm Irreducibility Gauss's Lemma Eisenstein's Criterion Reduction Modulo p Zeros of PolynomialsField Extensions Field Extensions Rational Expressions Simple ExtensionsSimple Extensions Algebraic and Transcendental Extensions The Minimal Polynomial Simple Algebraic Extensions Classifying Simple ExtensionsThe Degree of an ExtensionDefinition of the Degree The Tower LawRuler-and-Compass ConstructionsApproximate Constructions and More General Instruments Constructions in C Specific Constructions Impossibility Proofs Construction from a Given Set of PointsThe 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 IrrationalitiesNormality and Separability Splitting Fields Normality SeparabilityCounting Principles Linear Independence of MonomorphismsField Automorphisms K-Monomorphisms Normal ClosuresThe Galois CorrespondenceThe Fundamental Theorem of Galois TheoryA Worked ExampleSolubility and Simplicity Soluble Groups Simple Groups Cauchy's TheoremSolution by Radicals Radical Extensions An Insoluble Quintic Other MethodsAbstract 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 FieldsThe General Polynomial Equation Transcendence Degree Elementary Symmetric Polynomials The General Polynomial Cyclic Extensions Solving Equations of Degree Four or LessFinite Fields Structure of Finite Fields The Multiplicative Group Application to SolitaireRegular Polygons What Euclid Knew Which Constructions Are Possible? Regular Polygons Fermat Numbers How to Draw a Regular 17-gonCircle Division Genuine Radicals Fifth Roots Revisited Vandermonde Revisited The General Case Cyclotomic Polynomials Galois Group of Q( ) : 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 GroupAlgebraically 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 .