Classical Introduction to Galois Theory.
Title:
Classical Introduction to Galois Theory.
Author:
Newman, Stephen C.
ISBN:
9781118336670
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (298 pages)
Contents:
A CLASSICAL INTRODUCTION TO GALOIS THEORY -- CONTENTS -- PREFACE -- 1 CLASSICAL FORMULAS -- 1.1 Quadratic Polynomials -- 1.2 Cubic Polynomials -- 1.3 Quartic Polynomials -- 2 POLYNOMIALS AND FIELD THEORY -- 2.1 Divisibility -- 2.2 Algebraic Extensions -- 2.3 Degree of Extensions -- 2.4 Derivatives -- 2.5 Primitive Element Theorem -- 2.6 Isomorphism Extension Theorem and Splitting Fields -- 3 FUNDAMENTAL THEOREM ON SYMMETRIC POLYNOMIALS AND DISCRIMINANTS -- 3.1 Fundamental Theorem on Symmetric Polynomials -- 3.2 Fundamental Theorem on Symmetric Rational Functions -- 3.3 Some Identities Based on Elementary Symmetric Polynomials -- 3.4 Discriminants -- 3.5 Discriminants and Subfields of the Real Numbers -- 4 IRREDUCIBILITY AND FACTORIZATION -- 4.1 Irreducibility Over the Rational Numbers -- 4.2 Irreducibility and Splitting Fields -- 4.3 Factorization and Adjunction -- 5 ROOTS OF UNITY AND CYCLOTOMIC POLYNOMIALS -- 5.1 Roots of Unity -- 5.2 Cyclotomic Polynomials -- 6 RADICAL EXTENSIONS AND SOLVABILITY BY RADICALS -- 6.1 Basic Results on Radical Extensions -- 6.2 Gauss's Theorem on Cyclotomic Polynomials -- 6.3 Abel's Theorem on Radical Extensions -- 6.4 Polynomials of Prime Degree -- 7 GENERAL POLYNOMIALS AND THE BEGINNINGS OF GALOIS THEORY -- 7.1 General Polynomials -- 7.2 The Beginnings of Galois Theory -- 8 CLASSICAL GALOIS THEORY ACCORDING TO GALOIS -- 9 MODERN GALOIS THEORY -- 9.1 Galois Theory and Finite Extensions -- 9.2 Galois Theory and Splitting Fields -- 10 CYCLIC EXTENSIONS AND CYCLOTOMIC FIELDS -- 10.1 Cyclic Extensions -- 10.2 Cyclotomic Fields -- 11 GALOIS'S CRITERION FOR SOLVABILITY OF POLYNOMIALS BY RADICALS -- 12 POLYNOMIALS OF PRIME DEGREE -- 13 PERIODS OF ROOTS OF UNITY -- 14 DENESTING RADICALS -- 15 CLASSICAL FORMULAS REVISITED -- 15.1 General Quadratic Polynomial -- 15.2 General Cubic Polynomial.
15.3 General Quartic Polynomial -- APPENDIX A COSETS AND GROUP ACTIONS -- APPENDIX B CYCLIC GROUPS -- APPENDIX C SOLVABLE GROUPS -- APPENDIX D PERMUTATION GROUPS -- APPENDIX E FINITE FIELDS AND NUMBER THEORY -- APPENDIX F FURTHER READING -- REFERENCES -- INDEX.
Abstract:
Explore the foundations and modern applications of Galois theory Galois theory is widely regarded as one of the most elegant areas of mathematics. A Classical Introduction to Galois Theory develops the topic from a historical perspective, with an emphasis on the solvability of polynomials by radicals. The book provides a gradual transition from the computational methods typical of early literature on the subject to the more abstract approach that characterizes most contemporary expositions. The author provides an easily-accessible presentation of fundamental notions such as roots of unity, minimal polynomials, primitive elements, radical extensions, fixed fields, groups of automorphisms, and solvable series. As a result, their role in modern treatments of Galois theory is clearly illuminated for readers. Classical theorems by Abel, Galois, Gauss, Kronecker, Lagrange, and Ruffini are presented, and the power of Galois theory as both a theoretical and computational tool is illustrated through: A study of the solvability of polynomials of prime degree Development of the theory of periods of roots of unity Derivation of the classical formulas for solving general quadratic, cubic, and quartic polynomials by radicals Throughout the book, key theorems are proved in two ways, once using a classical approach and then again utilizing modern methods. Numerous worked examples showcase the discussed techniques, and background material on groups and fields is provided, supplying readers with a self-contained discussion of the topic. A Classical Introduction to Galois Theory is an excellent resource for courses on abstract algebra at the upper-undergraduate level. The book is also appealing to anyone interested in understanding the origins of Galois theory, why it was created, and how it has evolved into the discipline it is today.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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