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**Algebra. Volume II: Fields with structure, algebras and advanced topics. Transl. from the German by Silvio Levy. With the collaboration of the translator.**
*(English)*
Zbl 1130.12001

Universitext. New York, NY: Springer (ISBN 978-0-387-72487-4/pbk). ix, 336 p. (2008).

The book under review is the faithful English translation of the German textbook “Einführung in die Algebra: Teil II” by F. Lorenz [Mannheim etc.: B.I.-Wissenschaftsverlag (1990; Zbl 0727.12001)], and its second, slightly enlarged edition appeared seven years later (Zbl 0886.12001). Whereas the first part of this popular and well-established text for advanced undergraduate students (Zbl 0639.12008), the English translation of which became available a few months ago (Zbl 1087.12001), laid the foundations of field extensions, Galois theory, and their various applications in a didactically very original, profound and thorough manner, the current second volume is geared toward more seasoned students in the field of algebra. In fact, the author’s main guideline for the contents of this subsequent part is to present a greater variety of topics extending basic field and Galois theory in several directions, thereby providing both a wide panorama of more advanced applications and a broad choice of material for interested students and teachers.

With the main focus on fields with additional structure, semisimple algebras, and their applications in algebraic number theory, this volume offers another fifteen chapters following the nineteen chapters of the first volume (Zbl 1087.12001). Referring to the review of the German original (Zbl 0727.12001) as for further details, we recall here that the chapters of the present English edition of this second volume of F. Lorenz’s algebra text cover the following topics:

20. Ordered fields and real fields (including E. Artin’s characterization of real-closed fields and Sylvester’s theorem on the number of real roots);

21. Hilbert’s XVII. Problem and the real Nullstellensatz);

22. Orders and quadratic forms (via Witt rings);

23. Absolute values on fields (including non-Archimedean absolute values, the field of \(p\)-adic numbers, and Hensel’s lemma);

24. Residue class degree and ramification index of extensions;

25. Local fields;

26. Witt vectors, Witt’s lemma, and higher Artin-Schreier theory;

27. The Tsen rank of a field (including Tsen’s theorems, the Lang-Nagata theorem, the Chevalley-Warning theorem, and Krull’s dimension theorem);

28. Fundamentals of modules (simple and semisimple modules, Noetherian and Artinian modules; the Jordan-Hölder theorem, the Krull-Remak-Schmidt theorem, Artinian rings and algebras);

29. The Wedderburn theory of simple-algebras (incorporating Brauer groups of fields, tensor products of semisimple algebras, and the Skolem-Noether theorem);

30. Crossed products (with relative Brauer groups of Galois extensions, inflation and restriction maps, cyclic algebras, quaternion algebras, and cohomology of groups);

31. The Brauer group of a local field and the Hasse invariant;

32. Local class field theory (including the local norm residue symbol and its functoriality, the local reciprocity law, and the local Kronecker-Weber theorem);

33. Semisimple representations of finite groups (culminating in E. Artin’s induction theorem on characters and the Brauer-Witt induction theorem);

34. Schur algebras and the Schur group of a field.

As one can easily see, this is a large variety of specific (and fundamental) topics in advanced (field) algebra and algebraic number theory, which can be studied rather independently, according to the readers interest or taste. Such a wide spectrum of extra topics, especially in such an interesting and masterly composition, can barely be found in any other single textbook on advanced field theory. Moreover, these topics are by no means treated superficially. On the contrary, their diversity is coupled with an exposition in remarkable depth, with all the respective main theorems included.

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.

At the end of the book, there is an appendix of more than thirty pages, in which a large number of working problems and additional remarks is compiled. These refer to some of the chapters and come with a plentiful supply of hints and additional explanations, which must be seen as another particular feature of the book.

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.

With the main focus on fields with additional structure, semisimple algebras, and their applications in algebraic number theory, this volume offers another fifteen chapters following the nineteen chapters of the first volume (Zbl 1087.12001). Referring to the review of the German original (Zbl 0727.12001) as for further details, we recall here that the chapters of the present English edition of this second volume of F. Lorenz’s algebra text cover the following topics:

20. Ordered fields and real fields (including E. Artin’s characterization of real-closed fields and Sylvester’s theorem on the number of real roots);

21. Hilbert’s XVII. Problem and the real Nullstellensatz);

22. Orders and quadratic forms (via Witt rings);

23. Absolute values on fields (including non-Archimedean absolute values, the field of \(p\)-adic numbers, and Hensel’s lemma);

24. Residue class degree and ramification index of extensions;

25. Local fields;

26. Witt vectors, Witt’s lemma, and higher Artin-Schreier theory;

27. The Tsen rank of a field (including Tsen’s theorems, the Lang-Nagata theorem, the Chevalley-Warning theorem, and Krull’s dimension theorem);

28. Fundamentals of modules (simple and semisimple modules, Noetherian and Artinian modules; the Jordan-Hölder theorem, the Krull-Remak-Schmidt theorem, Artinian rings and algebras);

29. The Wedderburn theory of simple-algebras (incorporating Brauer groups of fields, tensor products of semisimple algebras, and the Skolem-Noether theorem);

30. Crossed products (with relative Brauer groups of Galois extensions, inflation and restriction maps, cyclic algebras, quaternion algebras, and cohomology of groups);

31. The Brauer group of a local field and the Hasse invariant;

32. Local class field theory (including the local norm residue symbol and its functoriality, the local reciprocity law, and the local Kronecker-Weber theorem);

33. Semisimple representations of finite groups (culminating in E. Artin’s induction theorem on characters and the Brauer-Witt induction theorem);

34. Schur algebras and the Schur group of a field.

As one can easily see, this is a large variety of specific (and fundamental) topics in advanced (field) algebra and algebraic number theory, which can be studied rather independently, according to the readers interest or taste. Such a wide spectrum of extra topics, especially in such an interesting and masterly composition, can barely be found in any other single textbook on advanced field theory. Moreover, these topics are by no means treated superficially. On the contrary, their diversity is coupled with an exposition in remarkable depth, with all the respective main theorems included.

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.

At the end of the book, there is an appendix of more than thirty pages, in which a large number of working problems and additional remarks is compiled. These refer to some of the chapters and come with a plentiful supply of hints and additional explanations, which must be seen as another particular feature of the book.

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.

Reviewer: Werner Kleinert (Berlin)

### MSC:

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |