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 Half-title......Page 3 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 Preface......Page 11 Introduction......Page 15 1. Polynomials, Polynomial Rings, Factorization, and Roots in C......Page 19 2.1. Approximating Roots......Page 26 2.2. Factoring Polynomials over Q......Page 27 2.4. Executing the Euclidean Algorithm over Q......Page 28 3. Ring Homomorphisms, Fields, Monomorphisms, and Automorphisms......Page 29 4. Groups, Permutations, and Permutation Actions......Page 31 5. Exercises......Page 32 6. The Property of Being Algebraic......Page 36 7. Minimal Polynomials......Page 37 8. The Field Generated by an Algebraic Number......Page 39 8.1. Rings and Vector Spaces Associated to an Algebraic Number......Page 40 8.2. The Ring Is a Field......Page 43 8.3. These Fields Are Isomorphic to Quotients of Polynomial Rings......Page 46 9. Reduced Forms in Q(α): Maple and Mathematica......Page 47 10. Exercises......Page 49 11.1. A Polynomial of Degree n Has at Most n Roots in Any Field Extension......Page 53 11.2. A Polynomial of Degree n Factors into n Linear Factors over C......Page 54 12. Which Algebraic Numbers Generate a Generated Field?......Page 56 12.1. Degrees of Minimal Polynomials of Algebraic Numbers Generating a Given Field......Page 57 12.2. If an Algebraic Number Generates a Field, So Do Its Affine Translations......Page 58 12.3. Degrees of Minimal Polynomials Divide the Dimension of an Enclosing Field......Page 59 12.4. The Set of Algebraic Numbers Is Closed Under Field Operations......Page 62 13. Exercise Set 1......Page 63 14. Computation in Algebraic Number Fields: Maple and Mathematica......Page 65 14.1. Declaring a Field......Page 66 14.2. Reduced Forms......Page 68 14.3. Factoring Polynomials over a Field......Page 69 14.5. The Euclidean Algorithm and Inverses......Page 70 14.6. Representing Algebraic Numbers and Finding Minimal Polynomials and Factors......Page 71 14.7. Reduced Forms over Subfields......Page 73 15. Exercise Set 2......Page 75 16. Fields Generated by Several Algebraic Numbers......Page 77 16.1. Generation by Two Algebraic Numbers Is Generation by One......Page 78 16.2. From Multiply Generated Extensions to Multivariate Polynomial Rings......Page 80 16.3. Fields Generated by a Finite Number of Algebraic Numbers Are Quotients of Polynomial Rings......Page 82 16.4. Splitting Fields......Page 85 17. Characterizing Isomorphisms between Fields: Three Cubic Examples......Page 86 18.1. Conditions for Isomorphisms from Multiply Generated Fields......Page 92 18.2. Isomorphisms of Splitting Fields over Isomorphic Fields......Page 95 19.1. Adjoining Arbitrarily Many Algebraic Numbers Leaves a Field Algebraic......Page 97 19.2. Properties of Characteristic Zero Fields: Simple versus Finite versus Algebraic Extensions......Page 98 20. Exercise Set 1......Page 100 21.1. Declaring a Field......Page 103 21.2. Declaring a Splitting Field......Page 106 21.3. Reduced Forms......Page 107 21.5. The Division Algorithm and Reduced Forms......Page 108 21.6. The Euclidean Algorithm and Inverses......Page 109 21.7. Representing Algebraic Numbers and Finding Minimal Polynomials and Factors......Page 110 21.8. Reduced Forms over Subfields......Page 111 21.9. Determining and Applying Automorphisms of Fields......Page 113 22. Exercise Set 2......Page 114 23. Normal Field Extensions and Splitting Fields......Page 117 24.1. Definition and Action......Page 119 24.2. The Order of the Galois Group Is the Dimension of the Field Extension......Page 120 24.3. Subfields Correspond to Subgroups and Vice Versa......Page 121 24.4. Subgroups Correspond to Subfields in a One-to-One Fashion......Page 123 24.5. Normal Subgroups Correspond to Splitting Fields......Page 125 25. Invariant Polynomials, Galois Resolvents, and the Discriminant......Page 129 25.1. Invariant Polynomials and Galois Resolvents......Page 131 25.2. Symmetric Polynomials and Resolvent Coefficients......Page 134 25.3. The Discriminant......Page 139 26. Exercise Set 1......Page 141 27. Distinguishing Numbers, Determining Groups......Page 142 27.1. Resolvent Factorization and Conjugacy......Page 144 27.2. Finding Subfields from Linear Factors......Page 146 27.3. Finding the Galois Group from Linear Factors......Page 149 28.1. The Galois Group, I......Page 151 28.2. Galois Resolvents......Page 155 28.3. The Galois Group, II......Page 157 28.4. Factorization and Resolvents......Page 162 29. Exercise Set 2......Page 163 30. Roots of Unity and Cyclotomic Extensions......Page 166 31. Cyclic Extensions over Fields with Roots of Unity......Page 170 32. Binomial Equations......Page 175 33. Ruler-and-Compass Constructions......Page 177 33.1. Ground Rules and Basic Constructions......Page 178 33.2. From Geometry to Field Theory......Page 179 33.3. From Field Theory to Impossibility Proofs......Page 181 34. Solvability by Radicals......Page 185 35. Characteristic p and Arbitrary Fields......Page 191 35.1. Basic Algebra and C......Page 192 35.2. Algebraic Elements, Field Extensions, andMinimal Polynomials......Page 193 35.3. Working with Algebraic Elements: Separability......Page 194 35.4. Multiply Generated Fields and the Galois Correspondence......Page 195 36. Finite Fields......Page 200 Historical Note......Page 207 1. The Subgroups of S4......Page 211 2. The Subgroups of S5......Page 212 Texts in advanced algebra and Galois theory......Page 215 Papers and monographs on Galois theory......Page 216 Related historic works......Page 217 Index......Page 219 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 How to understand the numbers we encountered in secondary school, and equations involving them: this is our point of departure in studying Galois theory.