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1. Classical Formulas 1
1.1 Quadratic Polynomials 4
1.2 Cubic Polynomials 6
1.3 Quartic Polynomials 10
2. Polynomials and Field Theory 14
2.1 Divisibility 15
2.2 Algebraic Extensions 24
2.3 Degree of Extensions 25
2.4 Derivatives 29
2.5 Primitive Element Theorem 31
2.6 Isomorphism Extension Theorem and Splitting Fields 36
3. Fundamental Theorem on Symmetric Polynomials and Discriminants 43
3.1 Fundamentals Theorem on Symmetric Polynomials 43
3.2 Fundamentals Theorem on Symmetric Rational Functions 50
3.3 Some Identities Based on Elementary Symmetric Polynomials 53
3. 4 Discriminants 56
3.5 Discriminants and Subfields of the Real Numbers 64
4. Irreducibility and Factorization 68
4.1 Irreducibility over the Rational Numbers 68
4.2 Irreducibility and Splitting Fields 72
4.3 Factorization and Adjunction 75
5. Roots of Unity and Cyclotomic Polynomials 84
5.1 Roots of Unity 84
5.2 Cyclotomic Polynomials 86
6. Radical Extensions and Solvability by Radicals 94
6.1 Basic Results on Radical Extensions 94
6.2 Gauss’s Theorem on Cyclotomic Polynomials 99
6.3 Abel’s Theorem on Radical Extensions 110
6.4 Polynomials of Prime Degree 116
7. General Polynomials and the Beginnings of Galois Theory 124
7.1 General Polynomials 124
7.2 The Beginnings of Galois Theory 131
8. Classical Galois Theory According to Galois 144
9. Modern Galois Theory 161
9.1 Galois Theory and Finite Extensions 163
9.2 Galois Theory and Splitting Fields 166
10. Cyclic Extensions and Cyclotomic Fields 183
10.1 Cyclic Extensions 183
10.2 Cyclotomic Fields 191
11. Galois’s Criterion for Solvability of Polynomials by Radicals 198
12. Polynomials of Prime Degree 206
13. Periods of Roots of Unity 215
14. Denesting Radicals 241
15. Classical Formulas Revisited 247
15.1 General Quadratic Polynomial 247
15.2 General Cubic Polynomial 249
15.3 General Quartic Polynomial 252
A. Cosets and Group Actions 262
B. Cyclic Groups 266
C. Solvable Groups 271
D. Permutation Groups 279
E. Finite Fields and Number Theory 288
F. Further Reading 292
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Add A Classical Introduction to Galois Theory, This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals. Both classical and modern approaches to the subject are described in turn in order to have the former (which is relatively c, A Classical Introduction to Galois Theory to the inventory that you are selling on WonderClubX
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Add A Classical Introduction to Galois Theory, This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals. Both classical and modern approaches to the subject are described in turn in order to have the former (which is relatively c, A Classical Introduction to Galois Theory to your collection on WonderClub |