Exploratory Galois Theory

Combining a concrete perspective with an exploration-based approach, Exploratory Galois Theory develops Galois theory at an entirely undergraduate level. The text grounds the presentation in the concept of algebraic numbers with complex approximations and assumes of its readers only a first course in abstract algebra. The author organizes the theory around natural questions about algebraic numbers, and exercises with hints and proof sketches encourage students' participation in the development. For readers with Maple or Mathematica, the text introduces tools for hands-on experimentation with finite extensions of the rational numbers, enabling a familiarity never before available to students of the subject. Exploratory Galois Theory includes classical applications, from ruler-and-compass constructions to solvability by radicals, and also outlines the generalization from subfields of the complex numbers to arbitrary fields. The text is appropriate for traditional lecture courses, for seminars, or for self-paced independent study by undergraduates and graduate students. Cover......Page 1 About......Page 2 Exploratory Galois Theory......Page 4 0521544998......Page 5 Contents......Page 8 Preface......Page 10 Introduction......Page 14 §1. Polynomials, Polynomial Rings, Factorization, and Roots in C......Page 18 §2. Computation with Roots and Factorizations: Maple and Mathematica......Page 25 §3. Ring Homomorphisms, Fields, Monomorphisms, and Automorphisms......Page 28 §4. Groups, Permutations, and Permutation Actions......Page 30 §5. Exercises......Page 31 §6. The Property of Being Algebraic......Page 35 §7. Minimal Polynomials......Page 36 §8. The Field Generated by an Algebraic Number......Page 38 §9. Reduced Forms in Q(a): Maple and Mathematica......Page 46 §10. Exercises......Page 48 §11. Minimal Polynomials Are Associated to Which Algebraic Numbers?......Page 52 §12. Which Algebraic Numbers Generate a Generated Field?......Page 55 §13. Exercise Set 1......Page 62 §14. Computation in Algebraic Number Fields: Maple and Mathematica......Page 64 §15. Exercise Set 2......Page 74 §16. Fields Generated by Several Algebraic Numbers......Page 76 §17. Characterizing Isomorphisms between Fields: Three Cubic Examples......Page 85 §18. Isomorphisms from Multiply Generated Fields......Page 91 §19. Fields and Splitting Fields Generated by Arbitrarily Many Algebraic Numbers......Page 96 §20. Exercise Set 1......Page 99 §21. Computation in Multiply Generated Fields: Maple and Mathematica......Page 102 §22. Exercise Set 2......Page 113 §23. Normal Field Extensions and Splitting Fields......Page 116 §24. The Galois Group......Page 118 §25. Invariant Polynomials, Galois Resolvents, and the Discriminant......Page 128 §26. Exercise Set 1......Page 140 §27. Distinguishing Numbers, Determining Groups......Page 141 §28. Computation of Galois Groups and Resolvents: Maple and Mathematica......Page 150 §29. Exercise Set 2......Page 162 §30. Roots of Unity and Cyclotomic Extensions......Page 165 §31. Cyclic Extensions over Fields with Roots of Unity......Page 169 §32. Binomial Equations......Page 174 §33. Ruler-and-Compass Constructions......Page 176 §34. Solvability by Radicals......Page 184 §35. Characteristic p and Arbitrary Fields......Page 190 §36. Finite Fields......Page 199 Historical Note......Page 206 §1. The Subgroups of S_4......Page 210 §2. The Subgroups of S_5......Page 211 Bibliography......Page 214 Index......Page 218 Combining a concrete perspective with an exploration-based approach, this analysis develops Galois theory at an entirely undergraduate level. The text grounds the presentation in the concept of algebraic numbers with complex approximations and only requires knowledge of a first course in abstract algebra. It introduces tools for hands-on experimentation with finite extensions of the rational numbers for readers with Maple or Mathematica. Please visit the author's website at: http://www.davidson.edu/academic/math/swallow/john.htm

Combining a concrete perspective with an exploration-based approach, Exploratory Galois Theory develops Galois theory at an entirely undergraduate level.

How to understand the numbers we encountered in secondary school, and equations involving them: this is our point of departure in studying Galois theory.