CiNii 図書 - Galois theory

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.

Factorization of polynomials
field extensions
the degree of an extension
ruler and compasses
transcendental numbers
the idea behind Galois theory
normality and separability
field degrees and group orders
monomorphisms, automorphisms and normal closures
the Galois correspondence
soluble and simple groups
solution of equations by radicals
the general polynomial equation
finite fields
regular polygons
calculating Galois groups
the fundamental theorem of algebra.

巻冊次: pbk. ISBN 9780412345500

内容説明

Galois theory is a fascinating mixture of classical and modern mathematics, and in fact provided much of the seed from which abstract algebra has grown. It is a showpiece of mathematical unification and of "technology transfer" to a range of modern applications. Galois Theory, Second Edition is a revision of a well-established and popular text. The author's treatment is rigorous, but motivated by discussion and examples. He further lightens the study with entertaining historical notes - including a detailed description of Evariste Galois' turbulent life. The application of the Galois group to the quintic equation stands as a central theme of the book. Other topics include the problems of trisecting the angle, duplicating the cube, squaring the circle, solving cubic and quartic equations, and the construction of regular polygons For this edition, the author added an introductory overview, a chapter on the calculation of Galois groups, further clarification of proofs, extra motivating examples, and modified exercises. Photographs from Galois' manuscripts and other illustrations enhance the engaging historical context offered in the first edition. Written in a lively, highly readable style while sacrificing nothing to mathematical rigor, Galois Theory remains accessible to intermediate undergraduate students and an outstanding introduction to some of the intriguing concepts of abstract algebra.

Preface to the First Edition Preface to the Second Edition Notes to the Reader Historical Introduction The Life of Galois Overview Background Factorization of Polynomials Field Extensions The Degree of an extension Ruler and Compasses Transcendental Numbers The Idea behind Galois Theory Normality and Separability Field Degrees and Group Order Monomorphisms, Automorphisms, and Normal Closures The Galois Correspondence A Specific Example Soluble and Simple Groups Solution of Equation by Radicals The General Polynomial Equation Finite Fields Regular Polygons Calculating Galois Groups The Fundamental Theorem of Algebra Selected Solutions References Index Symbol Index

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Galois theory

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