Elements of Algebra: Geometry, Numbers, Equations (Undergraduate Texts in Mathematics) Paperback – 19 Feb 2010
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"...The book is written in a very clear and pleasing style. Each of the 9 chapters of the book concludes with a section called "Discussions", which contains very interesting and valuable historical information and comments on the topics presented in the respective chapter. We strongly recommend this nice volume not only to beginners but also to experts."--MATHEMATICAL REVIEWS
From the Back Cover
This book is a concise, self-contained introduction to abstract algebra that stresses its unifying role in geometry and number theory. Classical results in these fields, such as the straightedge-and-compass constructions and their relation to Fermat primes, are used to motivate and illustrate algebraic techniques. Classical algebra itself is used to motivate the problem of solvability by radicals and its solution via Galois theory. This historical approach has at least two advantages: On the one hand it shows that abstract concepts have concrete roots, and on the other it demonstrates the power of new concepts to solve old problems. Algebra has a pedigree stretching back at least as far as Euclid, but today its connections with other parts of mathematics are often neglected or forgotten. By developing algebra out of classical number theory and geometry and reviving these connections, the author has made this book useful to beginners and experts alike. The lively style and clear exposition make it a pleasure to read and to learn from. --This text refers to the Hardcover edition.
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*) There appears to be a trend away from this attitude, but never as well executed as in Stillwell. The third edition of Stewart's Galois Theory is an example of how previously minimalist texts are being padded with lots of chitchat while little has changed in substance. The very mainstream algebra book of Beachy & Blair introduces the chapter on groups by saying that "the human body exhibits bilateral symmetry". There is none of this vague, pretentious nonsense in Stillwell.
Elements of Algebra is best studied along with Mathematics and Its History (Undergraduate Texts in Mathematics), also by Stillwell, which expands some of the topics that is treated too briefly in this book.
read enjoyable texts on such foundational topics, but reject the all-pervasive "Proposition/Proof/That's it, Figure-it-out-yourself/Proposition..." style on math books. For the rest of us. If "Elements of Number Theory" is recommended before diving into Dedekind's "Theory of Algebraic Integers", this one is great to prepare yourself for Emil Artin.
Thanks again, John.