| Description |
1 online resource (xii, 208 pages) |
| Physical Medium |
polychrome |
| Description |
text file |
| Bibliography |
Includes bibliographical references (pages 201-204) and index. |
| Summary |
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. |
| Contents |
Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Introduction -- CHAPTER ONE Preliminaries -- 1. Polynomials, Polynomial Rings, Factorization, and Roots in C -- 2. Computation with Roots and Factorizations: Maple and Mathematica -- 2.1. Approximating Roots -- 2.2. Factoring Polynomials over Q -- 2.3. Executing the Division Algorithm over Q -- 2.4. Executing the Euclidean Algorithm over Q -- 3. Ring Homomorphisms, Fields, Monomorphisms, and Automorphisms -- 4. Groups, Permutations, and Permutation Actions -- 5. Exercises. |
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CHAPTER TWO Algebraic Numbers, Field Extensions, and Minimal Polynomials -- 6. The Property of Being Algebraic -- 7. Minimal Polynomials -- 8. The Field Generated by an Algebraic Number -- 8.1. Rings and Vector Spaces Associated to an Algebraic Number -- 8.2. The Ring Is a Field -- 8.3. These Fields Are Isomorphic to Quotients of Polynomial Rings -- 9. Reduced Forms in Q(Ü): Maple and Mathematica -- 10. Exercises -- CHAPTER THREE Working with Algebraic Numbers, Field Extensions, and Minimal Polynomials -- 11. Minimal Polynomials Are Associated to Which Algebraic Numbers? |
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11.1. A Polynomial of Degree n Has at Most n Roots in Any Field Extension -- 11.2. A Polynomial of Degree n Factors into n Linear Factors over C -- 11.3. Minimal Polynomials Are Minimal Polynomials for n Distinct Algebraic Numbers -- 12. Which Algebraic Numbers Generate a Generated Field? -- 12.1. Degrees of Minimal Polynomials of Algebraic Numbers Generating a Given Field -- 12.2. If an Algebraic Number Generates a Field, So Do Its Affine Translations -- 12.3. Degrees of Minimal Polynomials Divide the Dimension of an Enclosing Field. |
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12.4. The Set of Algebraic Numbers Is Closed Under Field Operations -- 13. Exercise Set 1 -- 14. Computation in Algebraic Number Fields: Maple and Mathematica -- 14.1. Declaring a Field -- 14.2. Reduced Forms -- 14.3. Factoring Polynomials over a Field -- 14.4. The Division Algorithm and Reduced Forms -- 14.5. The Euclidean Algorithm and Inverses -- 14.6. Representing Algebraic Numbers and Finding Minimal Polynomials and Factors -- 14.7. Reduced Forms over Subfields -- 15. Exercise Set 2 -- CHAPTER FOUR Multiply Generated Fields -- 16. Fields Generated by Several Algebraic Numbers. |
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16.1. Generation by Two Algebraic Numbers Is Generation by One -- 16.2. From Multiply Generated Extensions to Multivariate Polynomial Rings -- 16.3. Fields Generated by a Finite Number of Algebraic Numbers Are Quotients of Polynomial Rings -- 16.4. Splitting Fields -- 17. Characterizing Isomorphisms between Fields: Three Cubic Examples -- 18. Isomorphisms from Multiply Generated Fields -- 18.1. Conditions for Isomorphisms from Multiply Generated Fields -- 18.2. Isomorphisms of Splitting Fields over Isomorphic Fields. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Galois theory.
|
|
Galois theory. |
| Genre/Form |
Electronic books.
|
| Other Form: |
Print version: Swallow, John, 1970- Exploratory Galois theory. Cambridge, UK ; New York : Cambridge University Press, 2004 9780521836500 (DLC) 2004045196 (OCoLC)54543749 |
| ISBN |
9780511648410 (electronic book) |
|
0511648413 (electronic book) |
|
9780511755200 (ebook) |
|
0511755201 (ebook) |
|
9780521836500 (hardback) |
|
0521836506 (hardback) |
|
1282394975 |
|
9781282394971 |
|
0521836506 (hardback) |
|
0521544998 (paperback) |
|
