Topics in Galois Theory

ISBN-13: 9781524960247 / Angielski / Twarda / 2019

Gallitano-Gupta

Spis treści

• 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 2?/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 ?
• 1. Transcendence of e
• 2. Transcendence of ?
• Chapter 9. A Radical Expression for cos 2?/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