Fields and Galois Theory: (Springer Undergraduate Mathematics Series)

Aimed at 3rd and 4th year undergraduates and beginning graduates, this book provides a gentle introduction to this popular subject. Assuming a background of a first course in abstract algebra, the book begins with a review of rings, ideals, quotients and homomorphisms. Polynomials, a key topic in field theory, are then introduced in the second chapter. Field extensions and splitting fields are the topics of Chapters 3 and 4, and there is an account of ruler and compass constructions, and a proof that "squaring the circle" is impossible, in Chapter 5. Chapter 6 uses the theory developed in Chapters 3 and 4 to give a description of finite fields, and includes a brief account of the use of such fields in coding theory. The book then concludes with the Galois group, normal and separable extensions, an account of polynomial equations, and the celebrated result that the quintic equation is not soluble by radicals.The aim is to provide a readable, "student-friendly" introduction that takes a more "natural" approach to its subject (as compared to the more formal introductions by Stewart and Garling), and that features clear explanations and plenty of worked examples and exercises - with full solutions - to encourage independent study.

Table of Contents:
Rings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.- Groups and Equations.- Regular Polygons.- Solutions.

Review :
From the reviews: "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 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) "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. ... 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 ... ." (Darren Glass, MathDL, February, 2006) "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)