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Short description 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. With exercises for each chapter, as well as useful appendices, this guide is ideal for anyone wanting a deeper appreciation of the origins of Galois theory, its fundamental concepts, and applications.
From the 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
