Fields and Galois Theory

1. Front Matter

   Pages i-x

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2. Rings and Fields

   Pages 1-24

3. Integral Domains and Polynomials

   Pages 25-49

4. Field Extensions

   Pages 51-69

5. Applications to Geometry

   Pages 71-78

6. Splitting Fields

   Pages 79-84

7. Finite Fields

   Pages 85-90

8. The Galois Group

   Pages 91-126

9. Equations and Groups

   Pages 127-147

10. Some Group Theory

    Pages 149-168

11. Groups and Equations

    Pages 169-181

12. Regular Polygons

    Pages 183-192

13. Solutions

    Pages 193-217

14. Back Matter

    Pages 219-225

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Fieldsaresetsinwhichallfouroftherationaloperations,memorablydescribed by the mathematician Lewis Carroll as “perdition, distraction, ugli?cation and derision”, can be carried out. They are assuredly the most natural of algebraic objects, since most of mathematics takes place in one ?eld or another, usually the rational ?eld Q, or the real ?eld R, or the complex ?eld C. This book sets out to exhibit the ways in which a systematic study of ?elds, while interesting in its own right, also throws light on several aspects of classical mathematics, notably on ancient geometrical problems such as “squaring the circle”, and on the solution of polynomial equations. The treatment is unashamedly unhistorical. When Galois and Abel dem- strated that a solution by radicals of a quintic equation is not possible, they dealt with permutations of roots. From sets of permutations closed under c- position came the idea of a permutation group, and only later the idea of an abstract group. In solving a long-standing problem of classical algebra, they laid the foundations of modern abstract algebra.

• Abstract algebra
• Field theory
• Galois theory
• Group theory
• Polynomials
• algebra
• finite field

From the reviews:

“This is a short but very good introductory book on abstract algebra, with emphasis on Galois Theory. Very little background in mathematics is required, so that the potential audience for this book range from undergraduate and graduate students, researchers, computer professionals, and the math enthusiasts.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, July, 2013)

"The author wrote this book to provide the reader with a treatment of classical Galois theory. … The book is well written. It contains many examples and over 100 exercises with solutions in the back of the book. Sprinkled throughout the book are interesting commentaries and historical comments. The book is suitable as a textbook for upper level undergraduate or beginning graduate students." (John N. Mordeson, Zentralblatt MATH, Vol. 1103 (5), 2007)

"To write such a book on a widely known but genuinely non-trivial topic is a challenge. … J. M. Howie did exactly what it takes. And he did it with such vigour and skill that the outcome is indeed absorbing and astounding. … Every paragraph has been scheduled with utmost care and the proofs are crystal clear. … the reader will never feel forlorn amidst brilliant theorems, which makes the book such a good read." (J. Lang, Internationale Mathematische Nachrichten, Issue 206, 2007)

"Howie’s book ... provides a rigorous and thorough introduction to Galois theory. ... this book would be an excellent choice for anyone with at least some backgound in abstract algebra who seeks an introduction to the study of Galois theory. Summing Up: Highly recommended. Upper-division undergraduates; graduate students." (D. S. Larson, CHOICE, Vol. 43 (10), June, 2006)

"The latest addition to Springer’s Undergraduate Mathematics Series is John Howie’s Fields and Galois Theory. … Howie is a fine writer, and the book is very self-contained. … Iknow 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 … ." (Darren Glass, MathDL, February, 2006)

"The book can serve as a useful introduction to the theory of fields and their extensions. The relevant background material on groups and rings is covered. The text is interspersed with many worked examples, as well as more than 100 exercises, for which solutions are provided at the end." (Chandan Singh Dalawat, Mathematical Reviews, Issue 2006 g)

• School of Mathematics and Statistics, University of St Andrews, North Haugh St Andrews Fife, UK

  John M. Howie

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