Galois theory (Chapman and Hall mathematics series)

TABLE OF CONTENTS: * Historical introduction -- * The life of Galois -- * Glossary of symbols -- 1.) Background -- 2.) Factorization of polynomials -- 3.) Field extensions -- 4.) The degree of an extension -- 5.) Ruler and compasses -- 6.) Transcendental numbers -- 7.) The idea behind Galois theory -- 8.) Normality and separability -- 9.) Field degrees and group orders -- 10.) Monomorphisms, automorphisms, and normal closures -- 11.) The Galois correspondence -- 12.) A specific example -- 13.) Some group theory -- 14.) Solution of equations by radicals -- 15.) The general polynomial equation -- 16.) Finite fields -- 17.) Regular polygons -- 18.) The 'fundamental theorem of algebra' -- 19.) Later developments -- 20.) Harder Exercises -- * Selected Solutions -- * References -- * INDEX . . . . . . . . . . This book is an attempt to present the Galois theory as a showpiece of mathematical unification, bringing together several different branches of the subject and creating a powerful machine for the study of problems of considerable historical and mathematical importance. The central theme is the application of the Galois group to the quintic equation. As well as the traditional approach by way of the 'general' polynomial equation, the author has included a direct approach which demonstrates the insolubility by radicals of a specific quintic polynomial with integer coefficients.