Graduate Texts in Mathematics. 158. New York, NY: Springer-Verlag. xii, 272 p. DM 48.00; öS 374.40; sFr 48.00 (1995).
This book is designed for a beginning graduate course in field theory. The reader should have some knowledge of basic group theory, ring theory, and field theory. In Chapters 0 and 1, the author lays down the material needed for an understanding of the remaining material on field theory. The author moves quickly through the first two chapters. However, he does an excellent job of stressing the key ideas. The remaining chapters on fields are packed with important material for the beginner in field theory. The book covers topics such as algebraic independence, Lüroth’s theorem, separable algebraic field extensions, Galois theory including the Krull topology, finite fields, Möbius inversion, roots of unity, Wedderburn’s theorem, and Kummer theory to name a few. In 272 pages, the author has gotten across many important ideas and results.
This book should not only work well as a textbook for a beginning graduate course in field theory, but also for a student who wishes to take a field theory course as independent study.