Topics in Galois Theory

Introduction and Acknowledgments
Notation

Chapter 1. Preliminaries
1. Algebraic Numbers
2. Algebraic Integers
3. Finite Fields

Chapter 2. Field Embedding and Extensions
1. Field Embedding
2. Some General Comments about Algebraic Numbers
3. Newton’s Identities

Chapter 3. Solution in Radicals
1. Solving a Cubic Equation (Cardano’s Formula)
2. Cubic with Real Coefficients
3. Radical Expressions for cos 2π/n
4. Gauss’ Construction of Regular 17-gon
5. Solution of the Quartic Equation

Chapter 4. Field Automorphisms and Galois Groups
1. Separable Elements and Separable Extensions
2. Some Properties of Normal Extensions
3. Automorphism of Fields—Galois Group
4. Fundamental Theorem of Galois Theory
5. Computing Galois Group of Polynomials over Q
6. Polynomial over Finite Fields
7. Automorphisms of Finite Fields
8. Irreducibility of Polynomials over Q

Chapter 5. Examples of Galois Groups and Galois Correspondence
1. The Galois Group of x³ − 2
2. The Galois Groups of x⁴ − 2
3. Galois Group of x⁸ − 24x⁶ + 144x⁴ − 288x² + 144
4. The Galois Group of x⁵ − 2
5. The Galois Group of x⁶ − 2
6. The Galois Group of x⁷ − 2
7. The Galois Group of x⁸ − 2
8. The Galois Group of x¹⁶ − 2

Chapter 6. Resultant and Discriminant
1. Discriminant of ax² + bx + c
2. Discriminant of x³ + px + q
3. Discriminant of Trinomial xⁿ + ax + b
4. The Polynomials fk(x), 2 ≤ k ≤ 5
5. The Galois Group of the Polynomial x⁷ − 154x + 99

Chapter 7. More Galois Groups
1. Polynomials with Cyclic Galois Group over the Q
2. Polynomials with the Symmetric Group Sn as a Galois Group

Chapter 8. Transcendence of e and π
1. Transcendence of e
2. Transcendence of π

Chapter 9. A Radical Expression for cos 2π/11

Chapter 10. A Natural Basis of Q(x) over Q(x^p)

Chapter 11. Some Interesting Irreducible Polynomials over Z
1. Theorem and Proof
2. Alternate Method
3. Examples

Appendix 1. Linear Independence of √p1, √p2, · · · , √pn over Q
Appendix 2. Ruler and Compass Construction
Appendix 3. Tate’s Proof of a Theorem of Dedekind
Appendix 4. About Solvable Quintics

Problems
Hints or Solutions to Problems
Glossary
Credits and Sources Acknowledged
Bibliography
Index