Topics in Galois Theory

AvGail Gallitano,Shiv Gupta

Inbunden, Engelska, 2019

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Produktinformation

Utgivningsdatum:2019-02-18
Format:Inbunden
Språk:Engelska
Antal sidor:277
Förlag:Kendall/Hunt Publishing Co ,U.S.
ISBN:9781524960247

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Innehållsförteckning

Introduction and AcknowledgmentsNotation Chapter 1. Preliminaries 1. Algebraic Numbers2. Algebraic Integers3. Finite FieldsChapter 2. Field Embedding and Extensions 1. Field Embedding2. Some General Comments about Algebraic Numbers3. Newton's IdentitiesChapter 3. Solution in Radicals 1. Solving a Cubic Equation (Cardano's Formula)2. Cubic with Real Coefficients3. Radical Expressions for cos 2?/n4. Gauss' Construction of Regular 17-gon5. Solution of the Quartic EquationChapter 4. Field Automorphisms and Galois Groups 1. Separable Elements and Separable Extensions2. Some Properties of Normal Extensions3. Automorphism of Fields—Galois Group4. Fundamental Theorem of Galois Theory5. Computing Galois Group of Polynomials over Q6. Polynomial over Finite Fields7. Automorphisms of Finite Fields8. Irreducibility of Polynomials over QChapter 5. Examples of Galois Groups and Galois Correspondence 1. The Galois Group of x3 ? 22. The Galois Groups of x4 ? 23. Galois Group of x8 ? 24x6 + 144x4 ? 288x2 + 1444. The Galois Group of x5 ? 25. The Galois Group of x6 ? 26. The Galois Group of x7 ? 27. The Galois Group of x8 ? 28. The Galois Group of x16 ? 2Chapter 6. Resultant and Discriminant 1. Discriminant of ax2 + bx + c2. Discriminant of x3 + px + q3. Discriminant of Trinomial xn + ax + b4. The Polynomials fk(x), 2 ? k ? 55. The Galois Group of the Polynomial x7 ? 154x + 99Chapter 7. More Galois Groups 1. Polynomials with Cyclic Galois Group over the Q2. Polynomials with the Symmetric Group Sn as a Galois GroupChapter 8. Transcendence of e and ? 1. Transcendence of e2. Transcendence of ?Chapter 9. A Radical Expression for cos 2?/11 Chapter 10. A Natural Basis of Q(x) over Q(xp) Chapter 11. Some Interesting Irreducible Polynomials over Z 1. Theorem and Proof2. Alternate Method3. ExamplesAppendix 1. Linear Independence of ?p1, ?p2, · · · , ?pn over QAppendix 2. Ruler and Compass ConstructionAppendix 3. Tate's Proof of a Theorem of DedekindAppendix 4. About Solvable QuinticsProblemsHints or Solutions to ProblemsGlossaryCredits and Sources AcknowledgedBibliographyIndex