A Classical Introduction to Galois Theory

Informationen zum Autor STEPHEN C. NEWMAN, MD, MSc, is Professor Emeritus of Psychiatry at the University of Alberta, Canada. He has published widely in psychiatric epidemiology and epidemiologic methods. Dr. Newman is the author of Biostatistical Methods in Epidemiology (Wiley). Klappentext Explore the foundations and modern applications of Galois theoryGalois 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 radicalsThroughout 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. Zusammenfassung With a focus on one central theme (the Impossibility Theorem) throughout, this highly accessible introduction to Galois theory presents a classical treatment of the topic and poses questions related to the solvability of polynomial equations by radicals. Modern points of view are also discussed in contrast to the historical development and context. Inhaltsverzeichnis Preface xi 1 Classical Formulas 1 1.1 Quadratic Polynomials 3 1.2 Cubic Polynomials 5 1.3 Quartic Polynomials 11 2 Polynomials and Field Theory 15 2.1 Divisibility 16 2.2 Algebraic Extensions 24 2.3 Degree of Extensions 25 2.4 Derivatives 29 2.5 Primitive Element Theorem 30 2.6 Isomorphism Extension Theorem and Splitting Fields 35 3 Fundamental Theorem on Symmetric Polynomials and Discriminants 41 3.1 Fundamental Theorem on Symmetric Polynomials 41 3.2 Fundamental Theorem on Symmetric Rational Functions 48 3.3 Some Identities Based on Elementary Symmetric Polynomials 50 3.4 Discriminants 53 3.5 Discriminants and Subfields of the Real Numbers 60 4 Irreducibility and Factorization 65 4.1 Irreducibility Over the Rational Numbers 65 4.2 Irreducibility and Splitting Fields 69 4.3 Factorization and Adjunction 72 5 Roots of Unity and Cyclotomic Polynomials 80 5.1 Roots of Unity 80 5.2 Cyclotomic Polynomials 82 6 Radical Extensions and Solvability by Radicals 89 6.1 Basic Results on Radical Extensions 89 6.2 Gauss's Theorem on Cyclotomic Polynomials 93 6.3 Abel's Theorem on Radical Extensions 104 6.4 Polynomials of Prime Degree 109 7 General Polynomials ...

List of contents

PREFACE xi
 
1 CLASSICAL FORMULAS 1
 
1.1 Quadratic Polynomials / 3
 
1.2 Cubic Polynomials / 5
 
1.3 Quartic Polynomials / 11
 
2 POLYNOMIALS AND FIELD THEORY 15
 
2.1 Divisibility / 16
 
2.2 Algebraic Extensions / 24
 
2.3 Degree of Extensions / 25
 
2.4 Derivatives / 29
 
2.5 Primitive Element Theorem / 30
 
2.6 Isomorphism Extension Theorem and Splitting Fields / 35
 
3 FUNDAMENTAL THEOREM ON SYMMETRIC POLYNOMIALS AND DISCRIMINANTS 41
 
3.1 Fundamental Theorem on Symmetric Polynomials / 41
 
3.2 Fundamental Theorem on Symmetric Rational Functions / 48
 
3.3 Some Identities Based on Elementary Symmetric Polynomials / 50
 
3.4 Discriminants / 53
 
3.5 Discriminants and Subfields of the Real Numbers / 60
 
4 IRREDUCIBILITY AND FACTORIZATION 65
 
4.1 Irreducibility Over the Rational Numbers / 65
 
4.2 Irreducibility and Splitting Fields / 69
 
4.3 Factorization and Adjunction / 72
 
5 ROOTS OF UNITY AND CYCLOTOMIC POLYNOMIALS 80
 
5.1 Roots of Unity / 80
 
5.2 Cyclotomic Polynomials / 82
 
6 RADICAL EXTENSIONS AND SOLVABILITY BY RADICALS 89
 
6.1 Basic Results on Radical Extensions / 89
 
6.2 Gauss's Theorem on Cyclotomic Polynomials / 93
 
6.3 Abel's Theorem on Radical Extensions / 104
 
6.4 Polynomials of Prime Degree / 109
 
7 GENERAL POLYNOMIALS AND THE BEGINNINGS OF GALOIS THEORY 117
 
7.1 General Polynomials / 117
 
7.2 The Beginnings of Galois Theory / 124
 
8 CLASSICAL GALOIS THEORY ACCORDING TO GALOIS 135
 
9 MODERN GALOIS THEORY 151
 
9.1 Galois Theory and Finite Extensions / 152
 
9.2 Galois Theory and Splitting Fields / 156
 
10 CYCLIC EXTENSIONS AND CYCLOTOMIC FIELDS 171
 
10.1 Cyclic Extensions / 171
 
10.2 Cyclotomic Fields / 179
 
11 GALOIS'S CRITERION FOR SOLVABILITY OF POLYNOMIALS BY RADICALS 185
 
12 POLYNOMIALS OF PRIME DEGREE 192
 
13 PERIODS OF ROOTS OF UNITY 200
 
14 DENESTING RADICALS 225
 
15 CLASSICAL FORMULAS REVISITED 231
 
15.1 General Quadratic Polynomial / 231
 
15.2 General Cubic Polynomial / 233
 
15.3 General Quartic Polynomial / 236
 
APPENDIX A COSETS AND GROUP ACTIONS 245
 
APPENDIX B CYCLIC GROUPS 249
 
APPENDIX C SOLVABLE GROUPS 254
 
APPENDIX D PERMUTATION GROUPS 261
 
APPENDIX E FINITE FIELDS AND NUMBER THEORY 270
 
APPENDIX F FURTHER READING 274
 
REFERENCES 277
 
INDEX 281

Product details

• Authors: Newman, Sc Newman, Stephen C Newman, Stephen C. Newman, Stephen C. (University of Alberta Newman, NEWMAN STEPHEN C, Newman Stephen C.
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 17.07.2012
• EAN: 9781118091395
• ISBN: 978-1-118-09139-5
• No. of pages: 296
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Mathematik, Algebra, Geschichte der Mathematik, Angewandte Mathematik, Mathematics, Applied mathematics, History of mathematics, Galoissche Theorie