Algebra Volume II: Fields with Structure, Algebras and Advanced Topics
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From the reviews: "The approach taken by the author is very straightforward, both in its presentation of proofs and its examples. Unlike some books, it is very possible to teach yourself the subject using this book. ... Out of all the books I used in undergrad, this one was one of my favorites because it was easy to use for reference and for independent learning. ... Great foundation establishing book to start building in the domain of Abstract Algebra." (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, May, 2014) "Algebra II: Fields with Structure, Algebras and Advanced Topics is ... a complete algebra course, including both undergraduate and graduate topics. ... The second volume focuses on fields with structure and algebras. ... the choice of topics and their organization are excellent and provide a unifying view of most of algebra. In all, Lorenz's book is a wonderful reference for both teachers and researches, and can be used with much profit for independent study by hard-working students." (Luiz Hendrique de Figueiredo, MathDL, July, 2008) "The author has managed to cover such an amazing wealth of advanced material in a very adroit manner, thereby keeping the representation utmost lively, comprehensible, thorough, always straight to the point, essentially self-contained, methodologically elegant and - all in all - admirably reader-friendly. ... No doubt, this is an outstanding textbook on advanced topics in abstract algebra, mainly in its field-theoretic and number-theoretic aspects, and its availability in English makes it a unique and utmost valuable enrichment of the international textbook literature on the subject." (Werner Kleinert, Zentralblatt MATH, Vol. 1130 (8), 2008)
The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra.
The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry.
The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, the Tsen rank of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory.
Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students.
From Reviews of the German version:
This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the
framework of the development of carefully selected problems.
- Stefan Porubsky, Mathematical Reviews