Synopses & Reviews
The book discusses classical Galois theory in considerable generality, treating fields of characteristic zero and of positive characteristic with consideration of both separable and inseparable extensions, but with a particular emphasis on algebraic extensions of the field of rational numbers. While most of the book is concerned with finite extensions, it discusses algebraic closure and infinite Galois extensions, and concludes with a new chapter on transcendental extensions. Key topics and features of this second edition: - Approaches Galois theory from the linear algebra point of view, following Artin; - Presents a number of applications of Galois theory, including symmetric functions, finite fields, cyclotomic fields, algebraic number fields, solvability of equations by radicals, and the impossibility of solution of the three geometric problems of Greek antiquity. Review from the first edition: "The text offers the standard material of classical field theory and Galois theory, though in a remarkably original, unconventional and comprehensive manner ... . the book under review must be seen as a highly welcome and valuable complement to existing textbook literature ... . It comes with its own features and advantages ... it surely is a perfect introduction to this evergreen subject. The numerous explaining remarks, hints, examples and applications are particularly commendable ... just as the outstanding clarity and fullness of the text." (Zentralblatt MATH, Vol. 1089 (15), 2006) Steven H. Weintraub is a Professor of Mathematics at Lehigh University and the author of seven books. This book grew out of a graduate course he taught at Lehigh. He is also the author of Algebra: An Approach via Module Theory (with W. A. Adkins).
Review
From the reviews: "The text offers the standard material of classical field theory and Galois theory, though in a remarkably original, unconventional and comprehensive manner ... . the book under review must be seen as a highly welcome and valuable complement to existing textbook literature ... . It comes with its own features and advantages ... it surely is a perfect introduction to this evergreen subject. The numerous explaining remarks, hints, examples and applications are particularly commendable ... just as the outstanding clarity and fullness of the text." (Werner Kleinert, Zentralblatt MATH, Vol. 1089 (15), 2006) From the reviews of the second edition: "The book is a valuable reference, covering many more topics than most of the standard books on the subject." (Mowaffaq Hajja, Zentralblatt MATH, Vol. 1195, 2010)
Synopsis
This is a textbook on Galois theory. Galois theory has a well-deserved re- tation as one of the most beautiful subjects in mathematics. I was seduced by its beauty into writing this book. I hope you will be seduced by its beauty in reading it. This book begins at the beginning. Indeed (and perhaps a little unusually for a mathematics text), it begins with an informal introductory chapter, Ch- ter 1. In this chapter we give a number of examples in Galois theory, even before our terms have been properly de?ned. (Needless to say, even though we proceed informally here, everything we say is absolutely correct.) These examples are sort of an airport beacon, shining a clear light at our destination as we navigate a course through the mathematical skies to get there. Then we start with our proper development of the subject, in Chapter 2. We assume no prior knowledge of ?eld theory on the part of the reader. We develop ?eld theory, with our goal being the Fundamental Theorem of Galois Theory (the FTGT). On the way, we consider extension ?elds, and deal with the notions of normal, separable, and Galois extensions. Then, in the penul- mate section of this chapter, we reach our main goal, the FTGT.
Synopsis
The new edition of this text on classical Galois Theory approaches the theory from the linear algebra point of view, following Artin. It also presents a number of applications of the theory and an expanded chapter on transcendental extensions.
Synopsis
Galois theory is a mature mathematical subject of particular beauty. Any Galois theory book written nowadays bears a great debt to Emil Artin's classic text "Galois Theory," and this book is no exception. While Artin's book pioneered an approach to Galois theory that relies heavily on linear algebra, this book's author takes the linear algebra emphasis even further. This special approach to the subject together with the clarity of its presentation, as well as the choice of topics covered, has made the first edition of this book a more than worthwhile addition to the literature on Galois Theory. The second edition, with a new chapter on transcendental extensions, will only further serve to make the book appreciated by and approachable to undergraduate and beginning graduate math majors.
About the Author
Steven H. Weintraub is a Professor of Mathematics at Lehigh University and author of seven books. This book grew out of a graduate course he taught at Lehigh. He is also the author of Algebra: An Approach via Module Theory (with W. A. Adkins).
Table of Contents
Introduction to Galois Theory.- Field Theory and Galois Theory.- Development and Applications of Galois Theory.- Extensions of the Field of Rational Numbers.- Further Topics in Field Theory.- Transcendental Extensions.- A. Some Results from Group Theory.- B. A Lemma on Constructing Fields.- C. A Lemma from Elementary Number Theory.- References.- Index.