Field Extensions and Galois Theory (Encyclopedia of Mathematics and its Applications, Series Number 22)

Originally published in 1984, the principal objective of this book is to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces. Galois theory is generally regarded as one of the central and most beautiful parts of algebra and its creation marked the culmination of investigations by generations of mathematicians on one of the oldest problems in algebra, the solvability of polynomial equations by radicals. 1.6.6. ExamplesPROBLEMS; 1.7. SEPARABLE POLYNOMIALS; 1.7.4. Examples; PROBLEMS; NOTES; Chapter 2 Algebraic Extensions; 2.1. ALGEBRAIC EXTENSIONS; 2.1.1. Examples; 2.1.7. Examples; PROBLEMS; 2.2. ALGEBRAICALLY CLOSED FIELDS; 2.2.8. Examples; PROBLEMS; 2.3. NORMAL EXTENSIONS; 2.3.1. Examples; 2.3.16. Examples; PROBLEMS; 2.4. PURELY INSEPARABLE EXTENSIONS; PROBLEMS; 2.5. SEPARABLE EXTENSIONS; PROBLEMS; NOTES; Chapter 3 Galois Theory; 3.1. SOME VECTOR SPACES OF MAPPINGS OF FIELDS; 3.1.4. Examples; PROBLEMS; 3.2. THE GENERAL GALOIS CORRESPONDENCES; 3.2.4. Examples; PROBLEMS; 3.3. GALOIS EXTENSIONS Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Editor's Statement; Foreword; Preface; Historical Introduction; Prerequisites; Notation; NOTATION PERTAINING TO THE THEORY OF FIELD EXTENSIONS AND GALOIS THEORY; Chapter 1 Preliminaries on Fields and Polynomials; 1.1. FIELDS OF FRACTIONS; PROBLEMS; 1.2. THE CHARACTERISTIC; 1.2.1. Examples; PROBLEMS; 1.3. PERFECT FIELDS AND PRIME FIELDS; PROBLEMS; 1.4. FIELD EXTENSIONS; 1.4.1. Examples; PROBLEMS; 1.5. FACTORIZATION OF POLYNOMIALS; 1.5.10. Examples; PROBLEMS; 1.6. SPLITTING OF POLYNOMIALS; 1.6.1. Examples; 1.6.2. Examples PROBLEMS3.4. FINITE GALOIS THEORY; PROBLEMS; 3.5. ROOTS OF UNITY; PROBLEMS; 3.6. PRIMITIVE ELEMENTS; PROBLEMS; 3.7. SEPARABLE AND INSEPARABLE DEGREES; PROBLEMS; 3.8. NORMS AND TRACES; 3.8.1. Examples; PROBLEMS; 3.9. CYCLIC EXTENSIONS; PROBLEMS; 3.10. SOLVABILITY BY RADICALS; PROBLEMS; 3.11. FINITE FIELDS; 3.11.4. Examples; PROBLEMS; 3.12. INFINITE GALOIS THEORY; PROBLEMS; NOTES; Suggestions for further reading; Chapter 4 Transcendental Extensions; 4.1. DIMENSIONAL OPERATORS; 4.1.1. Examples; 4.1.2. Examples; PROBLEMS; 4.2. TRANSCENDENCE BASES AND TRANSCENDENCE DEGREE; PROBLEMS 4.3. SPECIALIZATIONS AND PLACES OF FIELDS4.3.1. Examples; PROBLEMS; 4.4. SEPARABLE EXTENSIONS; PROBLEMS; 4.5. DERIVATIONS OF FIELDS; 4.5.1. Examples; 4.5.5. Examples; PROBLEMS; 4.6. DERIVATIONS OF ALGEBRAIC FUNCTION FIELDS; PROBLEMS; NOTES; Suggestions for further reading; References and Selected Bibliography; Bibliography; Of historical interest; Index This 1979 book shows how differential equation theory can be beautifully simplified by treating such equations from the product integral viewpoint. The first chapter, dealing with linear ordinary differential equations, is accessible to anyone with a knowledge of matrix theory and elementary calculus. Later chapters assume a more sophisticated reader. This 1984 book aims to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces. Galois theory is regarded amongst the central and most beautiful parts of algebra and its creation marked the culmination of generations of investigation.

this 1984 Book Makes The General Theory Of Field Extensions Accessible To Any Reader With A Modest Background In Groups, Rings And Vector Spaces.