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Fields and Galois Theory

John M. Howie
Publisher: 
Springer Verlag
Publication Date: 
2006
Number of Pages: 
225
Format: 
Paperback
Series: 
Springer Undergraduate Mathematics Series
Price: 
39.95
ISBN: 
1-85233-986-1
Category: 
Textbook
[Reviewed by
Darren Glass
, on
02/12/2006
]

The latest addition to Springer's Undergraduate Mathematics Series is John Howie's Fields and Galois Theory. As its title suggests, the book picks up where many undergraduate semester-long courses in Abstract Algbera would leave off — the author assumes familiarity with group theory, and the early chapters might be a bit rough to a reader who had never seen the definitions of rings and fields before — but quickly dives in to cover quite a few topics in the theory of fields.

The book opens with a pair of chapters covering the basics of rings, fields, and integral domains, including topics such as homomorphisms, unique factorization, and Eisenstein's criterion. Howie then moves on to discuss field extensions and their relationships with polynomials in two sections, appropriately entitled "Extensions and Polynomials" and "Polynomials and Extensions." The fourth chapter discusses the standard applications to geometry such as squaring the circle, but it does so in a much more concrete way than many books, which I think would be appreciated by many of our students.

The next three chapters treat splitting fields, finite fields, and Galois groups, and then the eighth chapter of the book pulls everything together by looking at approaches to solving quadratic, cubic, and quartic equations by radicals. Before giving the punchline that the quintic cannot be solved in this manner, Howie takes a detour through some group theory and either teaches or reminds the reader about solvable (or, as he calls them, "soluble") groups. The book then concludes with another nice and down-to-earth chapter about constructing regular polygons and how this geometric problem relates to Galois theory.

Howie is a fine writer, and the book is very self-contained. Throughout the book he mentions further generalizations of the theory (often asking how the situation would be different in characteristic p) and historical anecdotes, but keeps these as brief footnotes instead of expanding them fully. As a student, I know I preferred books that included the long digressions even if the professor didn't choose to cover them, but I know that many of my students would appreciate Howie's approach much more as it is not as overwhelming. This book also has a large number of good exercises, all of which have solutions in the back of the book. All in all, Howie has done a fine job writing a book on field theory — I am not sure if it is better than the other options out there, but it is certainly no worse.

Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include algebraic geometry, number theory, and (wait for it) Galois Theory. He can be reached at dglass@gettysburg.edu.

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1. Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Subrings, Ideals and Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 The Field of Fractions of an Integral Domain . . . . . . . . . . . . . . . . 13

1.4 The Characteristic of a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 A Reminder of Some Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . 20

2. Integral Domains and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Unique Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3. Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1 The Degree of an Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Extensions and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Polynomials and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4. Applications to Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.1 Ruler and Compasses Constructions . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 An Algebraic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5. Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

x Contents

6. Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7. The Galois Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.1 Monomorphisms between Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.2 Automorphisms, Groups and Subfields . . . . . . . . . . . . . . . . . . . . . . 94

7.3 Normal Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.4 Separable Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.5 The Galois Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.6 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.7 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8. Equations and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.1 Quadratics, Cubics and Quartics: Solution by Radicals . . . . . . . . 127

8.2 Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.3 Cyclic Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9. Some Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9.1 Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9.2 Sylow Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.3 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

9.4 Properties of Soluble Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

10. Groups and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

10.1 Insoluble Quintics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

10.2 General Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

10.3 Where Next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

11. Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

11.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

11.2 The Construction of Regular Polygons . . . . . . . . . . . . . . . . . . . . . . 187

12. Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Tags: 
Abstract Algebra
Field Theory
Galois Theory