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Abstract Algebra

Pierre Antoine Grillet
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 242
[Reviewed by
Miklós Bóna
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This book certainly has more than enough material for a two-semester algebra course for first-year graduate students. While the book is self-contained, its stlye is not for beginners; on that account it is necessary for students learning from this book to have had an undergraduate course on the topic first.

The balance among the various topics will please some instructors more than others. In particular, group theory is relegated to one-sixth of the book, and permutation groups are only treated in one section. There are universities where group theory constitutes half of the one-year algebra sequence; this book is not for them. Fields and rings are treated much more generously, in that they get three times the space that groups did. There are welcome chapters on topics that are not always part of a core course, such as lattices, algebras, functors, and Gröbner bases.

Some instructors who select this book will like the fact that no exercises have their solutions included in the book, making it possible to assign them for homework. Students will like this less, since they often benefit from studying fully worked-out solutions.

This reviewer would have liked to see end-of-chapter notes, explaining the "big picture", that is, applications of the material to other parts of mathematics, connections among various chapters, and a few intriguing open problems.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.

 Preface.- Groups.- Structure of Groups.- Rings.- Field Extensions.- Galois Theory.- Fields with Orders or Valuations.- Commutative Rings.- Modules.- Semisimple Rings and Modules.- Projectives and Injectives.- Constructions.- Ext and Tor.- Algebras.- Lattices.- Universal Algebra.-Categories.- Appendix.- References.- Further Readings.- Index.