A Course in Galois Theory

PREFACE
Part 1: Algebraic preliminaries
     1 Groups, fields and vector spaces
          1.1 Groups
          1.2 Fields
          1.3 Vector spaces
     2 The axiom of choice, and Zorn's lemma
          2.1 The axiom of choice
          2.2 Zorn's lemma
          2.3 The existence of a basis
     3 Rings
          3.1 Rings
          3.2 Integral domains
          3.3 Ideals
          3.4 Irreducibles, primes and unique factorization domains
          3.5 Principal ideal domains
          3.6 Highest common factors
          3.7 Polynomials over unique factorization domains
          3.8 The existence of maximal proper ideals
          3.9 More about fields
Part 2: The theory of fields, and Galois theory
     4 Field extensions
          4.1 Introduction
          4.2 Field extensions
          4.3 Algebraic and transcendental elements
          4.4 Algebraic extensions
          4.5 Monomorphisms of algebraic extensions
     5 Tests for irreducibility
          5.1 Introduction
          5.2 Eisenstein's criterion
          5.3 Other methods for establishing irreducibility
     6 Ruler-and-compass constructions
          6.1 Constructible points
          6.2 The angle pi/3 cannot be trisected
          6.3 Concluding remarks
     7 Splitting fields
          7.1 Splitting fields
          7.2 The extension of monomorphisms
          7.3 Some examples
     8 The algebraic closure of a field
          8.1 Introduction
          8.2 The existence of an algebraic closure
          8.3 The uniqueness of an algebraic closure
          8.4 Conclusions
     9 Normal extensions
          9.1 Basic properties
          9.2 Monomorphisms and automorphisms
     10 Separability
          10.1 Basic ideas
          10.2 Monomorphisms and automorphisms
          10.3 Galois extensions
          10.4 Differentiation
          10.5 The Frobenius monomorphism
          10.6 Inseparable polynomials
     11 Automorphisms and fixed fields
          11.1 Fixed fields and Galois groups
          11.2 The Galois group of a polynomial
          11.3 An example
          11.4 The fundamental theorem of Galois theory
          11.5 The theorem on natural irrationalities
     12 Finite fields
          12.1 A description of the finite fields
          12.2 An example
          12.3 Some abelian group theory
          12.4 The multiplicative group of a finite field
          12.5 The automorphism group of a finite field
     13 The theorem of the primitive element
          13.1 A criterion in terms of intermediate fields
          13.2 The theorem of the primitive element
          13.3 An example
     14 Cubics and quartics
          14.1 Extension by radicals
          14.2 The discriminant
          14.3 Cubic polynomials
          14.4 Quartic polynomials
     15 Roots of unity
          15.1 Cyclotomic polynomials
          15.2 Irreducibility
          15.3 The Galois group of a cyclotomic polynomial
     16 Cyclic extensions
          16.1 A necessary condition
          16.2 Abel's theorem
          16.3 A sufficient condition
          16.4 Kummer extensions
     17 Solution by radicals
          17.1 Soluble groups: examples
          17.2 Soluble groups: basic theory
          17.3 Polynomials with soluble Galois groups
          17.4 Polynomials which are solvable by radicals
     18 Transcendental elements and algebraic independence
          18.1 Transcendental elements and algebraic independence
          18.2 Transcendence bases
          18.3 Transcendence degree
          18.4 The tower law for transcendence degree
          18.5 Luroth's theorem
     19 Some further topics
          19.1 Generic polynomials
          19.2 The normal basis theorem
          19.3 Constructing regular polygons
     20 The calculation of Galois groups
          20.1 A procedure for determining the Galois group of a polynomial
          20.2 The soluble transitive subroups of Sigma_p
          20.3 The Galois group of a quintic
          20.4 Concluding remarks
Index
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