Algebra

From Math Reviews: This is Volume II of a two-volume introductory text in classical algebra. The text moves carefully with many details so that readers with some basic knowledge of algebra can read it without difficulty. The book can be recommended either as a textbook for some particular algebraic topic or as a reference book for consultations in a selected fundamental branch of algebra. The book contains a wealth of material. Amongst the topics covered in Volume II the reader can find: the theory of ordered fields (e.g., with reformulation of the fundamental theorem of algebra in terms of ordered fields, with Sylvester's theorem on the number of real roots), Nullstellen-theorems (e.g., with Artin's solution of Hilbert's 17th problem and Dubois' theorem), fundamentals of the theory of quadratic forms, of valuations, local fields and modules. The book also contains some lesser known or nontraditional results; for instance, Tsen's results on solubility of systems of polynomial equations with a sufficiently large number of indeterminates. These two volumes constitute a very good, readable and comprehensive survey of classical algebra and present a valuable contribution to the literature on this subject.

• algebra
• field
• finite group
• quadratic form
• matrix theory

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)

• University Mü nster, 48149, Germany

  Falko Lorenz

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