Foundations of Galois Theory

Foreword

Preface

I. The Elements of Galois Theory


1. The Elements of Field Theory


1. Preliminary Remarks


2. Some Important Types of Extensions


3. The Minimal Polynomial. The Structure of Simple Algebraic Extensions


4. The Algebraic Nature of Finite Extensions


5. The Structure of Composite Algebraic Extensions


6. Composite Finite Extensions


7. The Theorem That a Composite Algebraic Extension is Simple


8. The Field of Algebraic Numbers


9. The Composition of Fields


2. Necessary Facts from the Theory of Groups


1. The Definition of a Group


2. Subgroups, Normal Divisors and Factor Groups


3. Homomorphic Mappings


3. Galois Theory


1. Normal Extensions


2. Automorphisms of Fields. The Galois Group


3. The Order of the Galois Group


4. The Galois Correspondence


5. A Theorem About Conjugate Elements


6. The Galois Group of a Normal Subfield


7. The Galois Group of the Composition of Two Fields

II. The Solution of Equations by Radicals


1. Additional Facts from the General Theory of Groups


1. A Generalization of the Homomorphism Theorem


2. Normal Series


3. Cyclic Groups


4. Solvable and Abelian Groups


2. Equations Solvable by Radicals


1. Simple Radical Extensions


2. Cyclic Extensions


3. Radical Extensions


4. Normal Fields with Solvable Galois Group


5. Equations Solvable by Radicals


3. The Construction of Equations Solvable by Radicals


1. The Galois Group of an Equation as a Group of Permutations


2. The Factorization of Permutations into the Product of Cycles


3. Even Permutations. The Alternating Group


4. The Structure of the Alternating and Symmetric Groups


5. An Example of an Equation with Galois Group The Symmetric Group


6. A Discussion of the Results Obtained


4. The Unsolvability by Radicals of the General Equation of Degree N ≥ 5


1. The Field of Formal Power Series


2. The Field of Fractional Power Series


3. The Galois Group of the General Equation of Degree N


4. The Solution of Equations of Low Degree