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Fields and Galois Theory (Springer Undergraduate Mathematics Series) Paperback – September 26, 2007

ISBN-13: 978-1852339869 ISBN-10: 1852339861 Edition: 1st ed. 2005. Corr. 2nd printing 2007

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Product Details

  • Series: Springer Undergraduate Mathematics Series
  • Paperback: 226 pages
  • Publisher: Springer; 1st ed. 2005. Corr. 2nd printing 2007 edition (September 26, 2007)
  • Language: English
  • ISBN-10: 1852339861
  • ISBN-13: 978-1852339869
  • Product Dimensions: 7 x 0.6 x 10 inches
  • Shipping Weight: 13.6 ounces (View shipping rates and policies)
  • Average Customer Review: 3.8 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Best Sellers Rank: #1,878,164 in Books (See Top 100 in Books)

Editorial Reviews

Review

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. … 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 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)

From the Back Cover

The pioneering work of Abel and Galois in the early nineteenth century demonstrated that the long-standing quest for a solution of quintic equations by radicals was fruitless: no formula can be found. The techniques they used were, in the end, more important than the resolution of a somewhat esoteric problem, for they were the genesis of modern abstract algebra.

This book provides a gentle introduction to Galois theory suitable for third- and fourth-year undergraduates and beginning graduates. The approach is unashamedly unhistorical: it uses the language and techniques of abstract algebra to express complex arguments in contemporary terms. Thus the insolubility of the quintic by radicals is linked to the fact that the alternating group of degree 5 is simple - which is assuredly not the way Galois would have expressed the connection.

Topics covered include:

rings and fields

integral domains and polynomials

field extensions and splitting fields

applications to geometry

finite fields

the Galois group

equations

Group theory features in many of the arguments, and is fully explained in the text. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided.


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3.8 out of 5 stars
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Most Helpful Customer Reviews

2 of 2 people found the following review helpful By Christine on August 26, 2012
Format: Kindle Edition Verified Purchase
This text presents an introduction to rings and fields at an undergraduate level and corresponds to that portion of a course on abstract algebra that covers rings and fields. As with other titles in the S.U.M. series, this text has a nice selection of exercises accompanied by worked solutions, making the text helpful for unguided study.

The Kindle edition is presented using the mobi flowable text format rather than the print replica format and the result is quite disappointing as is too often the case with attempts to present mathematical material in eReader formats. The typesetting suffers from bad line wraps on inline formulae, orphaned headings for theorems, incorrect indenting of material and so on. All of these conspire to convert what is a nicely typeset work in physical print to an annoying and irritating eBook experience.
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5 of 7 people found the following review helpful By arpard fazakas on February 29, 2012
Format: Paperback Verified Purchase
Have you ever wondered why there are no general formulas for the roots of quintic or higher-degree polynomials with rational coefficients which involve only addition, subtraction, multiplication, division, and taking roots of the coefficents? Who hasn't? What, you say you haven't? Well, if you remember back to high-school algebra and the good old quadratic equation ax^2 + bx + c = 0, there is a solution x = -b plus or minus square root of b^2 minus 4ac, all over 2a. Turns out there is also a formula, albeit more complicated, for the cubic equation ax^3 + bx^2 + cx + d = 0, and an even more complicated one for the quartic equation ax^4 + bx^3 + cx^2 + dx + e = 0. But there's no such general formula for the roots of quintic (x^5) or higher polynomials with rational coefficients. The two mathematicians who proved this were both doomed to die at an early age: Niels Abel, a Norwegian who died of tuberculosis at age 27, and Evariste Galois, a Frenchman who died of a bullet in the gut received in a duel at age 20. (Ironically, no one remembers who or what they were fighting over. I hope it was worth it!)

If you'd like to know why this is so, this book will get you there. I've had a fair amount of exposure to higher math, so I'm not sure I can accurately determine whether someone with only high-school algebra could follow this book, but I think the answer would be a qualified "yes", assuming there was sufficient motivation and persistence. In addition to high-school algebra, the only other background someone would need is a minimum acquaintance with basic set theory, the barest minimum about complex numbers, and the beautiful Euler formula. The book explains almost everything beyond this that one would need to know, but does occasionally use technical terms which are not explicitly defined.
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1 of 1 people found the following review helpful By Ace on April 21, 2014
Format: Paperback Verified Purchase
I used this book in a second semester undergrad course in Abstract Algebra and I am not a fan of the organization of topics. Explanations of steps in between proofs and computations are sparse resulting in a hard book to follow.
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14 of 23 people found the following review helpful By Jorge Nakahara Jr. on February 24, 2006
Format: Paperback
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.
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3 of 6 people found the following review helpful By Amazon Customer on September 4, 2008
Format: Paperback Verified Purchase
I needed a book with examples of normal extensions. This one was very helpful.
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