Algebra. Volume I: Fields and Galois theory. Translated from the 1987 German edition by Silvio Levy. With the collaboration of the translator
This is the faithful English translation of the German textbook ``Einführung in die Algebra: Teil I'' by F. Lorenz, the first edition of which appeared in 1987 (Zbl 0639.12008). Due to its special character, this introduction to classical abstract algebra quickly became one of the most popular texts on the subject for advanced undergraduate students at German-language universities, and thus its second, slightly revised edition came out already five years later (1992). Now, almost twenty years after the appearance of the German original, the well-deserved English edition of this outstanding textbook has been brought about by Springer Verlag, with some further improvements and (finally) in modern printing. In view of the fact that there is certainly no lack of excellent introductions to topics of abstract algebra, it ought to be explained what the peculiar features of the textbook under review actually are. As the author points out in the preface, this text is to provide a lively, problem-oriented, comprehensible, but nevertheless profound introduction to those concepts and methods of modern abstract algebra that are related to field extensions and classical Galois theory, thereby assuming no more prerequisites than the basics of linear algebra. In this vein, the author sets a high value on both motivating concrete problems and detailed explanations of the theoretical framework for tackling them. Following this didactical principle, the author develops the conceptual tools gradually as needed, thereby avoiding the more or less usual, systematic, but somewhat dry standard style of many other, admittedly more encyclopedic textbooks on the subject. Despite its pronounced didactical touch, the text leads the reader right away into depth, and the author offers quite a number of interesting methodological approaches, extra topics, and arithmetical applications. As to the contents of this first volume of the author's introduction to algebra, the text consists of 19 sections treating the following subjects: (1) Constructibility with Ruler and Compass; (2) Algebraic Field Extensions; (3) Simple Extensions; (4) Fundamentals of Divisibility in Rings; (5) Prime Factorization in Polynomial Rings and Gauss's Theorem; (6) Splitting Fields; (7) Separable Extensions; (8) Galois Extensions; (9) Finite Fields, Cyclic Groups, and Roots of Unity; (10) Group Actions; (11) Applications of Galois Theory to Cyclotomic Fields; (12) Further Steps into Galois Theory; (13) Norms and Traces; (14) Binomial Equations; (15) Solvability of Algebraic Equations; (16) Integral Ring Extensions; (17) The Transcendence of \(\pi\); (18) Transcendental Field Extensions; (19) Hilbert's Nullstellensatz. As one can see from the table of contents, the author gradually introduces all those basics on fields, rings, groups, ideals, and algebras that are commonly touched upon in a first algebra course. Starting from prominent ancient problems in arithmetic and geometry, he simultaneously motivates and develops a good part of basic abstract algebra along its historical genesis, up to its present state of art, and even including the first steps into infinite Galois theory in Section 12, Dedekind's arithmetical reduction principle in Galois theory, and a remark on inverse Galois theory in Section 16. Furthermore, the Noether Normalization Theorem (Section 18) and Hilbert's Nullstellensatz (Section 19) not only provide striking applications of the core material, but also serve as appetizers for further studies with a view toward algebraic geometry. At the end of the book, there is an appendix containing a large number of exercises (often with hints) to the respective sections of the book as well as some additional remarks. The present English edition is enhanced by a carefully compiled index of notation. Now as before, this introduction to abstract algebra via fields and Galois theory is a didactic masterpiece, captivating the reader by its high degree of mathematical culture, individuality, liveliness, lucidity, and accuracy. It remains to wish that the English edition of this charming textbook will find the same enduring favour with students and teachers worldwide as the German original did, over nearly twenty years, in its local domain. The English translation of the second volume will appear in the near future. Its main focus will be on additional structures in field theory and related (arithmetic) topics, with much material not, covered in other introductory algebra textbooks, including real fields and quadratic forms, diophantine aspects, the calculus of Witt vectors, the Schur group of a field, elements of local class field theory, and other more advanced subjects. The international mathematical community may be looking forward to this valuable addition to the textbook literature in modern algebra, too.
scientific article; zbMATH DE number 5012869